Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections

K. Behrend^{1} and B. Fantechi^{2}

1 University of British Columbia, Vancouver, Canada. behrend@math.ubc.ca 2 SISSA, Trieste, Italy. fantechi@sissa.it

Summary. Let M and N be Lagrangian submanifolds of a complex symplectic manifold S. We construct a Gerstenhaber algebra structure on Tor_{∗}^{O_{S}} (O_{M}, O_{N}) and a compatible Batalin-Vilkovisky module structure on Ext_{O_{S}}^{∗} (O_{M}, O_{N}). This gives rise to a de Rham type cohomology theory for Lagrangian intersections.

Introduction

We are interested in intersections of Lagrangian submanifolds of holomorphic symplectic manifolds. Thus we work over the complex numbers in the analytic category.

There are two main aspects of this paper we would like to explain in the introduction: categorification of intersection numbers, and Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections.

Categorification of Lagrangian intersection numbers

This paper grew out of an attempt to categorify Lagrangian intersection numbers. We will explain what we mean by this, and how we propose a solution to the problem. Our construction looks very promising, but is still conjectural.

Lagrangian intersection numbers: smooth case

Let S be a (complex) symplectic manifold and L, M Lagrangian submanifolds. Since L and M are half-dimensional, the expected dimension of their intersection is zero. Intersection theory therefore gives us the intersection number

#(L ∩ M) ,

if the intersection is compact. In the general case, we get a class

[L ∩ M]^{vir} ∈ A_{0}(L ∩ M)

in degree zero Borel-Moore homology, such that in the compact case

#(L ∩ M) = deg[L ∩ M]^{vir} .

If the intersection X = L ∩ M is smooth,

[X]^{vir} = c_{⊤}(E) ∩ [X] ,

where E is the excess bundle of the intersection, which fits into the exact sequence

\xymatrix{0 \ar[r] & T_{X} \ar[r] & T_{L}|_{X} ⊕ T_{M}|_{X} \ar[r] & T_{S}|_{X} \ar[r] & E \ar[r] & 0}

of vector bundles on X. The symplectic form σ defines an isomorphism T_{S}|_{X} = Ω_{S}|_{X}. Under this isomorphism, the subbundle T_{L}|_{X} corresponds to the conormal bundle N_{L/S}^{∨}. Thus we can rewrite our exact sequence as

\xymatrix{0 \ar[r] & E^{∨} \ar[r] & N_{L/S}^{∨} ⊕ N_{M/S}^{∨} \ar[r] & Ω_{S}|_{X} \ar[r] & Ω_{X} \ar[r] & 0},

which shows that the excess bundle E is equal to the cotangent bundle Ω_{X}. Thus, in the smooth case,

[X]^{vir} = c_{⊤}(E) ∩ [X] = c_{⊤}(Ω_{X}) ∩ [X] = (−1)^{n}c_{⊤}(T_{X}) ∩ [X] ,

and in the smooth and compact case,

#(L ∩ M) = deg[X]^{vir} = (−1)^{n} ∫_{X} c_{⊤}(T_{X}) = (−1)^{n}χ(X) ,

where 2n is the dimension of S and χ(X) is the topological Euler characteristic of X. This shows that we can make sense of the intersection number even if the intersection is not compact: define the intersection number to be the signed Euler characteristic.

Intersection numbers: singular case

In [1], it was shown how to make sense of the statement that Lagrangian intersection numbers are signed Euler characteristics in the case that the intersection X is singular. An integer invariant ν_{X}(P) ∈ ℤ of the singularity of the analytic space X at the point P ∈ X was introduced.

In the case of a Lagrangian intersection X = L ∩ M, the number ν_{X}(P) can be described as follows. Locally around P, we can assume that S is equal to the cotangent bundle of M and M ⊂ S is the zero section. Moreover, we can assume that L is the graph of a closed, even exact, 1-form ω on M. If ω = df, for a holomorphic function f : M → ℂ, defined near P, then

ν_{X}(P) = (−1)^{n}(1 − χ(F_{P})) ,  (1)

where n = dim M and F_{P} is the Milnor fibre of f at P.

The main theorem of [1] implies that if L and M are Lagrangian submanifolds of the symplectic manifold S, with compact intersection X, then

#X = deg[X]^{vir} = χ(X, ν_{X}) ,

the weighted Euler characteristic of X with respect to the constructible function ν_{X}, which is defined as

χ(X, ν_{X}) = ∑_{i∈ℤ} i · χ({ν_{X} = i}) .

In particular, arbitrary Lagrangian intersection numbers are always welldefined: the intersection need not be smooth or compact. The integer ν_{X}(P) may be considered as the contribution of the point P to the intersection X = L ∩ M.

Categorifying intersection numbers: smooth case

To categorify the intersection number means to construct a cohomology theory such that the intersection number is equal to the alternating sum of Betti numbers. If X is smooth (not necessarily compact) a natural candidate is (shifted) holomorphic de Rham cohomology

#X = (−1)^{n}χ(X) = ∑(−1)^{i−n} dim_{ℂ} ℍ^{i}(X, (Ω_{X}^{•}, d)) .

Here (Ω_{X}^{•}, d) is the holomorphic de Rham complex of X and ℍ^{i} its hypercohomology. Of course, by the holomorphic Poincaré lemma, hypercohomology reduces to cohomology.

Categorification: compact case

If the intersection X = L ∩ M is compact, but not necessarily smooth, we have

#X = ∑_{i}(−1)^{i−n} dim_{ℂ} Ext_{O_{S}}^{i}(O_{L}, O_{M})

= ∑_{i,j}(−1)^{i}(−1)^{j−n} dim_{ℂ} H^{i}(X, Ext_{O_{S}}^{j}(O_{L}, O_{M})) .

If X is smooth, Ext_{O_{S}}^{j}(O_{L}, O_{M}) = Ω_{X}^{j}, so this reduces to Hodge cohomology

#X = ∑_{i,j}(−1)^{i}(−1)^{j−n} dim_{ℂ} H^{i}(X, Ω_{X}^{j}) .

This justifies using the sheaves Ext_{O_{S}}^{j}(O_{L}, O_{M}) as replacements for the sheaves Ω_{X}^{j} if X is not smooth any longer. To get finite-dimensional cohomology groups, we will construct de Rham type differentials

d : Ext_{O_{S}}^{j}(O_{L}, O_{M}) ⟶ Ext_{O_{S}}^{j+1}(O_{L}, O_{M}) ,

so that the hypercohomology groups

ℍ^{i}(X, (Ext_{O_{S}}^{•}(O_{L}, O_{M}), d))

are finite dimensional, even if X is not compact. Returning to the compact case, for any such d, we necessarily have

#X = ∑_{i}(−1)^{i−n} dim_{ℂ} ℍ^{i}(X, (Ext_{O_{S}}^{•}(O_{L}, O_{M}), d)) .

Categorification: local case

Every symplectic manifold S is locally isomorphic to the cotangent bundle Ω_{N} of a manifold N. The fibres of the induced vector bundle structure on S are Lagrangian submanifolds, and thus we have defined (locally on S) a foliation by Lagrangian submanifolds, i.e., a Lagrangian foliation. (Lagrangian foliations are also called polarizations.) We may assume that the leaves or our Lagrangian foliation of S are transverse to the two Lagrangians L and M whose intersection we wish to study. Then L and M turn into the graphs of 1-forms on N. The Lagrangian condition implies that these 1-forms on N are closed. Without loss of generality, we may assume that one of these 1-forms is the zero section of Ω_{N} and hence identify M with N. By making M = N smaller if necessary, we may assume that the closed 1-form defined by L is exact. Then L is the graph of the 1-form df, for a holomorphic function f on M. Thus the intersection L ∩ M is now the zero locus of the 1-form df:

X = Z(df) .

This is the local case.

Multiplying by df defines a differential

s : Ω_{M}^{j} ⟶ Ω_{M}^{j+1}
ω ⟼ df ∧ ω .

Because df is closed, the differential s commutes with the de Rham differential d : Ω_{M}^{j} → Ω_{M}^{j+1}. Thus the de Rham differential passes to cohomology with respect to s:

d : h^{j}(Ω_{M}^{•}, s) ⟶ h^{j+1}(Ω_{M}^{•}, s) ,

where h^{j} denotes the cohomology sheaves, which are coherent sheaves of O_{X}- modules. Let us denote these cohomology sheaves by

E^{j} = h^{j}(Ω_{M}^{•}, s) .

We have thus defined a complex of sheaves on X

(E^{•}, d) ,  (2)

where the E^{i} are coherent sheaves of O_{X}-modules, and the differential d is ℂlinear. It is a theorem of Kapranov [2], that the cohomology sheaves h^{i}(E^{•}, d) are constructible sheaves on X and thus have finite dimensional cohomology groups. It follows that the hypercohomology groups

ℍ^{i}(X, (E^{•}, d))

are finite-dimensional as well.

We conjecture that the constructible function

P ↦ ∑_{i}(−1)^{i−n} dim_{ℂ} ℍ_{{P }}^{i}(X, (E, d)) ,

of fiberwise Euler characteristic of (E, d) is equal to the function ν_{X}, from above (1). This would achieve the categorification in the local case. In particular, for the non-compact intersection numbers we would have

χ(X, ν_{X}) = ∑_{i}(−1)^{i−n} dim_{ℂ} ℍ^{i}(X, (E, d)) .

We remark that if f is a homogeneous polynomial (in a suitable set of coordinates), then this conjecture is true.

To make the connection with the compact case (and because this construction is of central importance to the paper), let us explain why

E^{i} = Ext_{O_{S}}^{i}(O_{L}, O_{M}) .

Denote the projection S = Ω_{M} → M by π. The 1-form on Ω_{M} which corresponds to the vector field generating the natural ℂ^{∗}-action on the fibres we shall call α. Then dα = σ is the symplectic form on S. We consider the 1-form s = α − π^{∗}df on S. Its zero locus in S is equal to the graph of df. Let us denote the subbundle of Ω_{S} annihilating vector fields tangent to the fibres of π by E. Then s ∈ Ω_{S} is a section of E and we obtain a resolution of the structure sheaf of O_{L} over O_{S}:

\xymatrix{… \ar[r] & Λ^{2}E^{∨} \ar[r]^{s̃} & E^{∨} \ar[r]^{s̃} & O_{S}} ,

where s̃ denotes the derivation of the differential graded O_{S}-algebra Λ^{•}E^{∨} given by contraction with s. Taking duals and tensoring with O_{M}, we obtain a complex of vector bundles (ΛE|_{M}, s|_{M}) which computes Ext_{O_{S}}^{i}(O_{L}, O_{M}). One checks that (ΛE|_{M}, s|_{M}) = (Ω_{M}, s).

Categorification: global case

We now come to the contents of this paper. let S be a symplectic manifold and L, M Lagrangian submanifolds with intersection X. Let us use the abbreviation E^{i} = Ext_{O_{S}}^{i}(O_{L}, O_{M}). The E^{i} are coherent sheaves of O_{X}-modules. The main theorem of this paper is that the locally defined de Rham differentials (2) do not depend on the way we write S as a cotangent bundle, or, in other words, that d is independent of the chosen polarization of S. Thus, the locally defined d glue, and we obtain a globally defined canonical de Rham type differential

d : E^{i} → E^{i+1} .

In the case that X is smooth, E^{i} = Ω_{X}^{i}, and d is the usual de Rham differential. We may call (E^{•}, d) the virtual de Rham complex of the Lagrangian intersection X. Conjecturally, (E, d) categorifies Lagrangian intersection numbers in the sense that for the local contribution of the point P ∈ X to the Lagrangian intersection we have

ν_{X}(P) = ∑_{i}(−1)^{i−n} dim_{ℂ} ℍ_{{P }}^{i}(X, (E, d)) .

Hence, for the non-compact intersection numbers we should have

χ(X, ν_{X}) = ∑_{i}(−1)^{i−n} dim_{ℂ} ℍ^{i}(X, (E, d)) .

In particular, if the intersection is compact, #X = χ(X, ν_{X}) should be the alternating sum of the Betti numbers of the hypercohomology groups of the virtual de Rham complex.

Donaldson-Thomas invariants

Our original motivation for this research was a better understanding of Donaldson-Thomas invariants. It is to be hoped that the moduli spaces giving rise to Donaldson-Thomas invariants (spaces of stable sheaves of fixed determinant on Calabi-Yau threefolds) are Lagrangian intersections, at least locally. We have two reasons for believing this: first of all, the obstruction theory giving rise to the virtual fundamental class is symmetric, a property shared by the obstruction theories of Lagrangian intersections. Secondly, at least heuristically, these moduli spaces are equal to the critical set of the holomorphic Chern-Simons functional.

Our ‘exchange property’ should be useful for gluing virtual de Rham complexes if the moduli spaces are only local Lagrangian intersections.

In this way we hope to construct a virtual de Rham complex on the Donaldson-Thomas moduli spaces and thus categorify Donaldson-Thomas invariants.

Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections

The virtual de Rham complex (E^{•}, d) is just one half of the story. There is also the graded sheaf of O_{X}-algebras A^{•} given by

A^{i} = Tor_{−i}^{O_{S}}(O_{L}, O_{M}) ,

Locally, A^{•} is given as the cohomology of (ΛT_{M}, s̃), in the above notation. The Lie-Schouten-Nijenhuis bracket induces a ℂ-linear bracket operation

[ ,] : A • ⊗_{ℂ}A^{•} ⟶ A^{•}

of degree +1. We show that these locally defined brackets glue to give a globally defined bracket making (A^{•}, ∧, [ ]) a sheaf of Gerstenhaber algebras.

Then E^{•} is a sheaf of modules over A^{•}. (The module structure is induced by contraction.) The bracket on A^{•} and the differential on E^{•} satisfy a compatibility condition, see (5). We say that (E, d) is a Batalin-Vilkovisky module over the Gerstenhaber algebra (A, ∧, [ ]). (This structure has been called a calculus by Tamarkin and Tsygan in [4].)

In the case that L and M are oriented submanifolds, i.e., the highest exterior powers of the normal bundles have been trivialized, we have an identification

A^{i} = E^{n+i} .

Transporting the differential from E^{•} to A^{•} via this identification turns (A, ∧, [ ], d) into a Batalin-Vilkovisky algebra.

To prove these facts we have to study differential Gerstenhaber algebras and differential Batalin-Vilkovisky modules over them. We will prove that locally defined differential Gerstenhaber algebras and their differential Batalin-Vilkovisky modules are quasi-isomorphic, making their cohomologies isomorphic and hence yielding the well-definedness of the bracket and the differential.

First order truncation

In this paper we are only interested in the Gerstenhaber and Batalin-Vilkovisky structures on A and E. In other words, we only deal with the structures induced on cohomology. This amounts to a truncation of the full derived Lagrangian intersection. Because of our modest goal, we only need to study differential Gerstenhaber and Batalin-Vilkovisky structures up to first order. In future research, we hope to address the complete derived structure on Lagrangian intersections.

This would certainly involve studying the Witten deformation of the de Rham complex in more detail. Related work along these lines has been done by Kashiwara and Schapira [3].

Overview

1. Algebra

In this introductory section, we discuss algebraic preliminaries. We review the definitions of differential Gerstenhaber algebra, and differential Batalin-Vilkovisky module. This is mainly to fix our notation. There are quite a few definitions to keep track of; we apologize for the lengthiness of this section.

2. Symplectic geometry

Here we review a few basic facts about complex symplectic manifolds. In particular, the notions of Lagrangian foliation, polarization, and the canonical partial connection are introduced.

3. Derived Lagrangian intersections on polarized symplectic manifolds

On a polarized symplectic manifold, we define derived intersections of Lagrangian submanifolds. These are (sheaves of) Gerstenhaber algebras on the scheme theoretic intersection of two Lagrangian submanifolds. The main theorem we prove about these derived intersections is a certain invariance property with respect to symplectic correspondences. We call it the exchange property.

We repeat this program for derived homs, (the Batalin-Vilkovisky case), and oriented derived intersections (the oriented Batalin-Vilkovisky case).

4. The Gerstenhaber structure on Tor and the Batalin-Vilkovisky structure on Ext

In this section we use the exchange property to prove that, after passing to cohomology, we do not notice the polarization any more. The Gerstenhaber and Batalin-Vilkovisky structures are independent of the polarization chosen to define them.

This section closes with an example of a symplectic correspondence and the corresponding exchange property.

5. Further remarks

In this final section we define virtual de Rham cohomology of Lagrangian intersections. We speculate on what virtual Hodge theory might look like. We introduce a natural differential graded category associated to a complex symplectic manifold. (It looks like a kind of holomorphic, de Rham type analogue of the Fukaya category.) Finally, we mention the conjectures connecting the virtual de Rham complex to the perverse sheaf of vanishing cycles.

1 Algebra

Let M be a manifold. Regular functions, elements of O_{M}, have degree 0. By ΛT_{M} we mean the graded sheaf of polyvector fields on M. We think of it as a sheaf of graded O_{M}-algebras (the product being ∧), concentrated in non-positive degrees, the vector fields having degree −1. By Ω_{M}^{•} we denote the graded sheaf of differential forms on M. This we think of as a sheaf of graded O_{M}-modules, concentrated in non-negative degrees, with 1-forms having degree +1. We will denote the natural pairing of T_{M} with Ω_{M} by X ⌟ ω ∈ O_{S}, for X ∈ T_{M} and ω ∈ Ω_{M}. The following is, of course, well known:

Lemma 1.1. There exists a unique extension of ⌟ to an action of the sheaf of graded O_{M}-algebras ΛT_{M} on the sheaf of graded O_{S}-modules Ω_{M}^{•}, which satisfies

(i) f ⌟ ω = fω, for f ∈ O_{S} and ω ∈ Ω_{M}^{•} (linearity over O_{M}), (ii) X ⌟ (ω_{1} ∧ ω_{2}) = (X ⌟ ω_{1}) ∧ ω_{2} + (−1)^{\overline{ω}^{1}}ω_{1} ∧ (X ⌟ ω_{2}), for X ∈ T_{M} and

ω_{1}, ω_{2} ∈ Ω_{M}^{•}, (the degree −1 part acts by derivations), (iii) (X ∧ Y) ⌟ ω = X ⌟ (Y ⌟ ω), for X, Y ∈ ΛT_{M}, ω ∈ Ω_{M}^{•} (action property).

Now turn things around and note that any section s ∈ Ω_{M} defines a derivation of degree +1 on ΛT_{M}, which we shall denote by s̃. It is the unique derivation which extends the map T_{M} → O_{M} given by s̃(X) = X ⌟ s, for all X ∈ T_{M}. (Note that this is not a violation of the universal sign convention, see Remark 1.3.)

Lemma 1.2. The pair (ΛT_{M}, s̃) is a sheaf of differential graded O_{M}-algebras. Left multiplication by s defines a differential on Ω_{M}^{•} and the pair (Ω_{M}^{•}, s) is a sheaf of differential graded modules over (ΛT_{M}, s̃).

Proof. This amounts to the formula

s ∧ (X ⌟ ω) = s̃(X) ⌟ ω + (−1)\overline{X}X ⌟ (s ∧ ω)  (3)

for all ω ∈ Ω_{M}^{•} and X ∈ ΛT_{M}. ⊓ ⊔

Remark 1.3. Set ⟨X, ω⟩ equal to the degree zero part of X ⌟ ω. This is a perfect pairing ΛT_{M} ⊗_{O_{M}} Ω_{M}^{•} → O_{M}, expressing the fact that Ω_{M}^{•} is the O_{M}- dual of ΛT_{M}. According to Formula (3), we have, if deg X + deg ω + 1 = 0,

⟨s̃(X), ω⟩ + (−1)\overline{X}⟨X, s ∧ ω⟩ = 0 .

This means that the derivation s̃ and left multiplication by s are O_{S}-duals of one another. To explain the signs, note that we think of s̃ and s as differentials on the graded sheaves ΛT_{M} and Ω_{M}^{•}, and for differentials of degree +1 the sign convention is

0 = D⟨X, ω⟩ = ⟨DX, ω⟩ + (−1)\overline{X}⟨X, Dω⟩ .

In particular, for deg X = 1 and ω = 1 we get s̃(X) = ⟨X, s⟩ = X ⌟ s.

Remark 1.4. We can summarize Formula (3) more succinctly as

[s, i_{X}] = i_{s̃(X)} ,

where i_{X} : Ω_{M}^{•} → Ω_{M}^{•} denotes the endomorphism ω ↦ X ⌟ ω.

1.1 Differential Gerstenhaber algebras

Let S be a manifold and A a graded sheaf of O_{S}-modules.

Definition 1.5. A bracket on A of degree +1 is a homomorphism

[ ] : A ⊗_{ℂ} A ⟶ A

of degree +1 satisfying:

(i) [ ] is a graded ℂ-linear derivation in each of its two arguments, (ii) [ ] is graded commutative (not anti-commutative). If [ ] satisfies, in addition, the Jacobi identity, we shall call [ ] a Lie bracket.

The sign convention for brackets of degree +1 is that the comma is treated as carrying the degree +1, the opening and closing bracket as having degree 0. Thus, when passing an odd element past the comma, the sign changes. For example, the graded commutativity reads:

[Y, X] = (−1)^{\overline{X}\overline{Y} +\overline{X}+\overline{Y}} [X, Y] .

Definition 1.6. A Gerstenhaber algebra over O_{S} is a sheaf of graded O_{S}- modules A, concentrated in non-positive degrees, endowed with

(i) a commutative (associative, of course) product ∧ of degree 0 with unit, making A a sheaf of graded O_{S}-algebras,

(ii) a Lie bracket [ ] of degree +1 (see Definition 1.5).

In our cases, the underlying O_{S}-module of A will always be coherent and O_{S} → A^{0} will be a surjection of coherent O_{S}-algebras. The main example is the following:

Example 1.7. Let M ⊂ S be a submanifold and A = Λ_{O_{M}} T_{M} the polyvector fields on M. The bracket is the Schouten-Nijenhuis bracket.

Definition 1.8. A differential Gerstenhaber algebra is a Gerstenhaber algebra A over O_{S} endowed with an additional ℂ-linear map s̃ : A → A of degree +1 which satisfies

(i) [s̃, s̃] = s^{2} = 0, (ii) s̃ is a derivation with respect to ∧, in particular it is O_{S}-linear, (iii) s̃ is a derivation with respect to [ ].

Thus, neglecting the bracket, a differential Gerstenhaber algebra is a sheaf of differential graded algebras over O_{S}.

Lemma 1.9. Let (A, s̃) be a differential Gerstenhaber algebra. Let I ⊂ A^{0} be the image of s̃ : A^{−1} → A^{0}. This is a sheaf of ideals in A^{0}. Then the cohomology h^{∗}(A, s̃) is a Gerstenhaber algebra with h^{0}(A, s̃) = A^{0}/I.

Proof. This is clear: the fact that s̃ is a derivation with respect to both products on A implies that the two products pass to h^{∗}(A, s̃). Then all the properties of the products pass to cohomology. ⊓ ⊔

Example 1.10. Let M ⊂ S and A = ΛT_{M} be as in Example 1.7. In addition, let s ∈ Ω_{M} be a closed 1-form. Then (ΛT_{M}, s̃) with ∧ and Schouten-Nijenhuis bracket [ ] is a differential Gerstenhaber -algebra. The closedness of s makes s̃ a derivation with respect to [ ].

1.2 Morphisms of differential Gerstenhaber algebras

Definition 1.11. Let A and B be Gerstenhaber algebras over O_{S}. A morphism of Gerstenhaber algebras is a homomorphism ϕ : A → B of graded O_{S}-modules (of degree zero) which is compatible with both ∧ and [ ]:

(i) ϕ(X ∧ Y) = ϕ(X) ∧ ϕ(Y), (ii) ϕ([X, Y]) = [ϕ(X), ϕ(Y)].

Definition 1.12. Let (A, s̃) and (B, t̃) be differential Gerstenhaber algebras over O_{S}. A (first order) morphism of differential Gerstenhaber algebras is a pair (ϕ, { }), where ϕ : A → B is a degree zero homomorphism of graded O_{S}-modules, and { } : A ⊗_{ℂ} A → B is a degree zero ℂ-bilinear map , such that

(i) ϕ(X ∧ Y) = ϕ(X) ∧ ϕ(Y) and ϕ(s̃X) = t̃ϕ(X), so that ϕ : A → B is a morphisms of differential graded O_{S}-algebras,

(ii) \{ \} is symmetric, i.e., \{Y, X\} = (−1)^{\overline{X}\overline{Y}} \{X, Y \}, (iii) { } is a ℂ-linear derivation with respect to ∧ in each of its arguments,

where the A-module structure on B is given by ϕ, in other words,

\{X ∧ Y, Z\} = ϕ(X) ∧ \{Y, Z\} + (−1)^{\overline{X}\overline{Y}} ϕ(Y) ∧ \{X, Z\} ,

and

\{X, Y ∧ Z\} = \{X, Y \} ∧ ϕ(Z) + (−1)^{\overline{Y} \overline{Z}}\{X, Z\} ∧ ϕ(Y) ,

(iv) the default of ϕ to commute with [ ] is equal to the default of the O_{S}-linear differentials to behave as derivations with respect to { },

ϕ[X, Y]−[ϕ(X), ϕ(Y)] = (−1)\overline{X}t̃\{X, Y \}−(−1)\overline{X}\{s̃X, Y \}−\{X, s̃Y \} . (4)

Remark 1.13. We will always omit the qualifier ‘first order’, as we will not consider any ‘higher order’ morphisms in this paper. This is because, in the end, we are only interested in the cohomology of our differential Gerstenhaber algebras. To keep track of the induced structure on cohomology, first order morphisms suffice. We hope to return to ‘higher order’ questions in future research.

Remark 1.14. Suppose all conditions in Definition 1.12 except the last are satisfied. Then both sides of the equation in Condition (iv) are symmetric of degree one and ℂ-linear derivations with respect to ∧ in each of the two arguments. Thus, to check Condition (iv), it suffices to check on ℂ-algebra generators for A.

Lemma 1.15. A morphism of differential Gerstenhaber algebras

(ϕ, { }) : (A, s̃) ⟶ (B, t̃)

induces a morphism of Gerstenhaber algebras on cohomology. In other words,

h^{∗}(ϕ) : h^{∗}(A, s̃) ⟶ h^{∗}(B, t̃)

respects both ∧ and [ ].

Proof. Any morphism of differential graded O_{S}-algebras induces a morphism of graded algebras when passing to cohomology. Thus h^{∗}(ϕ) respects ∧. The fact that h^{∗}(ϕ) respects the Lie brackets, follows form Property (iv) of Definition 1.12. All three terms on the right hand side of said equation vanish in cohomology. ⊓ ⊔

Definition 1.16. A quasi-isomorphism of differential Gerstenhaber algebras is a morphism of differential Gerstenhaber algebras which induces an isomorphism of Gerstenhaber algebras on cohomology.

1.3 Differential Batalin-Vilkovisky modules

Definition 1.17. Let A be a Gerstenhaber algebra. A sheaf of graded O_{S}- modules L, with an action ⌟ of A, making L a graded A-module, is called a Batalin-Vilkovisky module over A, if it is endowed with a ℂ-linear map d : L → L of degree +1 satisfying

(i) [d, d] = d^{2} = 0, (ii) For all X, Y ∈ A and every ω ∈ L we have

d(X ∧ Y ⌟ ω) + (−1)^{\overline{X}+\overline{Y}} X ∧ Y ⌟ dω + (−1)\overline{X}[X, Y] ⌟ ω =

(−1)\overline{X}X ⌟ d(Y ⌟ ω) + (−1)^{\overline{X}\overline{Y} +\overline{Y}} Y ⌟ d(X ⌟ ω) . (5)

Remark 1.18. Write i_{X} for the endomorphism ω ↦ X ⌟ ω of L. Then Formula (5) can be rewritten as

[X, Y] ⌟ ω = [[i_{X}, d], i_{Y} ](ω)

or simply

i_{[X,Y]} = [[i_{X}, d], i_{Y} ] .  (6)

Note also that [[i_{X}, d], i_{Y} ] = [i_{X}, [d, i_{Y}]].

The action property (X ∧ Y) ⌟ ω = X ⌟ (Y ⌟ ω) translates into i_{X∧Y} = i_{X} ∘ i_{Y}.

In our applications, Batalin-Vilkovisky modules will always be coherent over O_{S}. Note that there is no multiplicative structure on L, so there is no requirement for the differential d to be a derivation.

Example 1.19. Let M ⊂ S and A = ΛT_{M} be the Gerstenhaber algebra of polyvector fields on M, as in Example 1.7. Then Ω_{M}^{•} with exterior differentiation d is a Batalin-Vilkovisky module over ΛT_{M}.

Definition 1.20. A differential Batalin-Vilkovisky module over the differential Gerstenhaber algebra (A, s̃) is a Batalin-Vilkovisky module L for the underlying Gerstenhaber algebra A, endowed with an additional ℂ-linear map s : L → L of degree +1 satisfying:

(i) [s, s] = s^{2} = 0, (ii) (M, s) is a differential graded module over the differential graded algebra

(A, s̃), i.e., we have

s(X ⌟ ω) = s̃(X) ⌟ ω + (−1)\overline{X}X ⌟ s(ω) ,

for all X ∈ A, ω ∈ L. More succinctly: [s, i_{X}] = i_{s̃(X)}. (iii) [d, s] = 0.

Note that the differential s is necessarily O_{S}-linear. This distinguishes it from d.

Lemma 1.21. Let (L, s) be a differential Batalin-Vilkovisky module over the differential Gerstenhaber algebra (A, s̃). Then h^{∗}(L, s) is a Batalin-Vilkovisky module for the Gerstenhaber algebra h^{∗}(A, s̃).

Proof. First, h^{∗}(M, s) is a graded h^{∗}(A, s̃)-module. The condition [d, s] = 0 implies that d passes to cohomology. Then the properties of d pass to cohomology as well. ⊓ ⊔

Example 1.22. Let M ⊂ S be a submanifold and s ∈ Ω_{M} a closed 1-form. Then (Ω_{M}^{•}, s) (see Lemma 1.2) is a differential Batalin-Vilkovisky module over the differential Gerstenhaber algebra (ΛT_{M}, s̃) of Example 1.10.

1.4 Homomorphisms of differential Batalin-Vilkovisky modules

Definition 1.23. Let A and B be Gerstenhaber algebras and ϕ : A → B a morphism of Gerstenhaber algebras. Let L be a Batalin-Vilkovisky module over A and M a Batalin-Vilkovisky module over B. A homomorphism of Batalin-Vilkovisky modules of degree n (covering ϕ) is a degree n homomorphism of graded A-modules ψ : L → M (where the A-module structure on M is defined via ϕ), which commutes with d:

(i) ψ(X ⌟ ω) = (−1)^{n\overline{X}}ϕ(X) ⌟ ψ(ω), (ii) ψd_{L}(ω) = (−1)^{n}d_{M}ψ(ω). We write the latter condition as [ψ, d] = 0.

Definition 1.24. Let (A, s̃) and (B, t̃) be differential Gerstenhaber algebras and (ϕ, { }) : (A, s̃) → (B, t̃) a morphism of differential Gerstenhaber algebras. Let (L, s) be a differential Batalin-Vilkovisky module over (A, s̃) and (M, t) a differential Batalin-Vilkovisky module over (B, t̃). A (first order) homomorphism of differential Batalin-Vilkovisky modules of degree n covering (ϕ, { }) is a pair (ψ, δ), where ψ : (L, s) → (M, t) is a degree n homomorphism of differential graded (A, s̃)-modules, where the (A, s̃)-module structure on (M, t) is through ϕ. Moreover, δ : L → M is a ℂ-linear map, also of degree n, satisfying

(i) the commutator property

ψ ∘ d − (−1)^{n}d ∘ ψ = −2(−1)^{n}t ∘ δ + 2δ ∘ s ,  (7)

(ii) compatibility with the bracket { } property

δ(X ∧Y ⌟ ω)+(−1)^{n(\overline{X}+\overline{Y})}ϕ(X)∧ϕ(Y) ⌟ δω+(−1)^{n(\overline{X}+\overline{Y})}\{X, Y \} ⌟ ψ(ω)

= (−1)^{n\overline{X}}ϕ(X) ⌟ δ(Y ⌟ ω) + (−1)^{\overline{X}\overline{Y} +n\overline{Y}} ϕ(Y) ⌟ δ(X ⌟ ω) . (8)

Remark 1.25. The same comments as those in Remark 1.13 apply.

Remark 1.26. If we use the same letter s to denote the O_{S}-linear differentials on L and M, we can rewrite the commutator conditions of Definition 1.24 more succinctly as

[ψ, s] = 0 [ψ, d] − 2[δ, s] = 0 .

The compatibility with bracket property can be rewritten as

[ι_{X}, [ι_{Y} , δ]] = ι_{{X,Y }} ∘ ψ .  (9)

Note the absence of a condition on the commutator [δ, d]. This would be a ‘higher order’ condition.

Remark 1.27. It is a formal consequence of properties of the commutator bracket that the left hand side of (9) is a ℂ-linear derivation in each of its two arguments X, Y. The same is true of the right hand side by assumption. Thus we have: if all properties of Definition 1.24, except for (i) and (ii) are satisfied, then to check that (ii) is satisfied, it suffices to do this for all X and Y belonging to a set of ℂ-algebra generators for A.

Remark 1.28. Suppose all properties of Definition 1.24 except for (i) are satisfied. Suppose also that L is free of rank one as an A-module on the basis ω^{∘} ∈ L. Then it suffices to prove Equation (7) applied to elements of the form X ⌟ ω^{0}, where X runs over a set of generators of A as an A^{0}-module.

Lemma 1.29. Let (ψ, δ) : (L, s) → (M, t) be a homomorphism of differential Batalin-Vilkovisky modules over the morphism (ϕ, { }) : (A, s̃) → (B, t̃) of differential Gerstenhaber algebras. Then h^{∗}(ψ) : h^{∗}(L, s) → h^{∗}(M, t) is a homomorphism of Batalin-Vilkovisky modules over the morphism of Gerstenhaber algebras h^{∗}(ϕ) : h^{∗}(A, s̃) → h^{∗}(B, t̃).

Proof. Evaluating the right hand side of Equation 7 on s-cocycles in L, yields t-boundaries in M. ⊓ ⊔

1.5 Invertible differential Batalin-Vilkovisky modules

Definition 1.30. We call the Batalin-Vilkovisky module L over the Gerstenhaber algebra A invertible, if, locally in S, there exists a section ω^{∘} of L such that the evaluation homomorphism

Ψ^{∘} : A ⟶ L
X ⟼ (−1)^{\overline{X}\overline{ω}^{∘}}X ⌟ ω^{∘}

is an isomorphism of sheaves of O_{S}-modules. Any such ω^{∘} will be called a (local) orientation for L over A.

Note that if the degree of an orientation ω^{∘} is n, then L^{k} = 0, for all k > n, by our assumption on A. Thus orientations always live in the top degree of L. Moreover, if orientations exist everywhere locally, L^{n} is an invertible sheaf over A^{0}.

Lemma 1.31. Let L be an invertible Batalin-Vilkovisky module over the Gerstenhaber algebra A and assume that ω^{∘} is a (global) orientation for L over A. Then, transporting the differential d via Ψ^{∘} to A yields a ℂ-linear map of degree +1 which we will call d^{∘} : A → A. It is characterized by the formula

d^{∘}(X) ⌟ ω^{∘} = d(X ⌟ ω^{∘}) .

It squares to 0 and it satisfies:

(−1)\overline{X}[X, Y] = d^{∘}(X) ∧ Y + (−1)\overline{X}X ∧ d^{∘}(Y) − d^{∘}(X ∧ Y) , (10)

for all X, Y ∈ A. In other words, d^{∘} is a generator for the bracket [ ], making A a Batalin-Vilkovisky algebra.

Proof. Follows directly from Formula (5) upon noticing that because ω^{∘} is top-dimensional, it is automatically d-closed: dω^{∘} = 0. ⊓ ⊔

Corollary 1.32. If the Gerstenhaber algebra admits an invertible Batalin-Vilkovisky module it is locally a Batalin-Vilkovisky algebra.

Example 1.33. The Batalin-Vilkovisky module Ω_{M}^{•} over the Gerstenhaber algebra ΛT_{M} of Example1.10 is invertible. Any non-vanishing top-degree form ω^{∘} ∈ Ω_{M}^{n} is an orientation for Ω_{M}^{•}, where n = dim M. Thus, the Schouten-Nijenhuis algebra ΛT_{M} is a Batalin-Vilkovisky algebra. For Calabi-Yau manifolds, i.e., Ω_{M}^{n} = O_{S}, a generator for the Batalin-Vilkovisky algebra is given.

Definition 1.34. Let (L, s) be a differential Batalin-Vilkovisky module over the differential Gerstenhaber algebra (A, s̃). Then (L, s) is called invertible, if the underlying Batalin-Vilkovisky module L is invertible over the underlying Gerstenhaber algebra A. An orientation for (A, s̃) is an orientation of the underlying L.

Proposition 1.35. Let (L, s) be an invertible differential Batalin-Vilkovisky module over the differential Gerstenhaber algebra (A, s̃). Then under the isomorphism Ψ^{∘} defined by an orientation ω^{∘} of L over A, the differential s̃ corresponds to the differential s. In particular, the induced differential d^{∘} on A has the property

[d^{∘}, s̃] = 0 ,

besides satisfying (10). Hence (A, d^{∘}, s̃) is a differential Batalin-Vilkovisky algebra.

Moreover, the cohomology h^{∗}(L, s) is an invertible Batalin-Vilkovisky module over the Gerstenhaber algebra h^{∗}(A, s̃). We have h^{n}(L, s) = L^{n}/I and the image of any orientation of L over A under the quotient map L^{n} → L^{n}/I gives an orientation for h^{∗}(L, s) over h^{∗}(A, s̃).

Proof. The equation s ∘ Ψ^{∘} = (−1)^{\overline{ω}^{∘}}Ψ^{∘} ∘ s̃ follows immediately from [s, i_{X}] = i_{s̃(X)} upon noticing that s(ω) = 0. As Ψ^{∘} is therefore an isomorphism of differential graded O_{S}-modules, the cohomology is isomorphic: h^{∗}(A, s̃) \underrightarrow{∼} h^{∗}(L, s). The rest follows from this. ⊓ ⊔

Example 1.36. For a closed 1-form s on M, the differential Batalin-Vilkovisky module (Ω_{M}^{•}, s) over the differential Gerstenhaber algebra (ΛT_{M}, s̃) of Example 1.22 is invertible. Any trivialization of Ω_{M}^{n} defines an orientation.

1.6 Oriented homomorphisms of invertible Batalin-Vilkovisky modules

Definition 1.37. Let ϕ : A → B be a morphism of Gerstenhaber algebras and ψ : L → M a homomorphism of invertible Batalin-Vilkovisky modules covering ϕ. Let ω_{L}^{∘} and ω_{M}^{∘} be orientations for L and M, respectively. The homomorphism ψ : L → M is said to preserve the orientations (or be oriented) if ψ(ω_{L}^{∘}) = ω_{M}^{∘}.

Lemma 1.38. Suppose given oriented invertible Batalin-Vilkovisky modules L and M over the Gerstenhaber algebras A and B, making A and B into Batalin-Vilkovisky algebras. Suppose ψ : L → M is an oriented homomorphism of Batalin-Vilkovisky modules. Then under the identifications of L and M with A and B given by ω_{L}^{∘} and ω_{M}^{∘}, the map ψ : L → M corresponds to ϕ : A → B. Hence ϕ : A → B commutes with d^{∘}. Thus ϕ is a morphism of Batalin-Vilkovisky algebras: it respects ∧, [ ] and d^{∘}.

Definition 1.39. Let (ψ, δ) : (L, s) → (M, t) be a homomorphism of invertible differentiable Batalin-Vilkovisky modules over (ϕ, { }) : (A, s̃) → (B, t̃). Let ω_{L}^{∘} and ω_{M}^{∘} be orientations for L and M, respectively. We call (ψ, δ) oriented if ψ(ω_{L}^{∘}) = ω_{M}^{∘} and δ(ω_{L}^{∘}) = 0.

Proposition 1.40. Suppose (ψ, δ) : (L, s, ω_{L}^{∘}) → (M, t, ω_{M}^{∘}) is an oriented homomorphism of oriented invertible differential Batalin-Vilkovisky modules over (ϕ, { }) : (A, s̃) → (B, t̃). Then (A, s̃, [ ], d^{∘}) and (B, t̃, [ ], d^{∘}) are differential Batalin-Vilkovisky algebras. Transporting δ : L → M via the identifications of L and M with A and B to a map δ^{∘} : A → B, satisfying

δ^{∘}(X) ⌟ ω_{M} = (−1)^{\overline{δ}\overline{X}}δ(X ⌟ ω_{L}) ,

we get a triple

(ϕ, { }, δ^{∘}) : (A, s̃, [ ], d^{∘}) ⟶ (B, t̃, [ ], d^{∘}) ,

which satisfies the following conditions:

(i) ϕ : (A, s̃) → (B, t̃) is a morphism of differential graded algebras, (ii) the commutator property

ϕ ∘ d^{∘} − d^{∘} ∘ ϕ = −2 t̃∘ δ^{∘} + 2 δ^{∘} ∘ s̃,

or, by abuse of notation, [ϕ, d^{∘}] − 2[δ^{∘}, s̃] = 0, (iii) the map δ^{∘} is a generator for the bracket { },

{X, Y } = δ^{∘}(X) ∧ ϕ(Y) + ϕ(X) ∧ δ^{∘}(Y) − δ^{∘}(X ∧ Y) ,

(iv) the default of ϕ to preserve [ ] equals the default of s̃ to be a derivation with respect to { }, Equation (4).

Thus (ϕ, { }, δ^{∘}) is a (first order) morphism of differential Batalin-Vilkovisky algebras.

The Lie bracket [ ] is determined by its generator d^{0}, and the bracket { } is determined by its generator δ^{0}. Thus, in a certain sense, the two brackets are redundant. Moreover, Condition (iv) is implied by Conditions (ii) and (iii).

Remark 1.41. A morphism of differential Batalin-Vilkovisky algebras

(ϕ, { }, δ^{∘}) : (A, s̃, [ ], d^{∘}) ⟶ (B, t̃, [ ], d^{∘})

induces on cohomology

h^{∗}(ϕ) : (h^{∗}(A, s̃), [ ], d^{∘}) → (h^{∗}(B, t̃), [ ], d^{∘})

a morphism of Batalin-Vilkovisky algebras.

2 Symplectic geometry

Let (S, σ) be a symplectic manifold, i.e., a complex manifold S endowed with a closed holomorphic 2-form σ ∈ Ω_{S}^{2} which is everywhere non-degenerate, i.e., X → X ⌟ σ defines an isomorphism of vector bundles T_{S} → Ω_{S}. The (complex) dimension of S is even, and we will denote it by 2n.

A submanifold M ⊂ S is Lagrangian, if the restriction of this isomorphism T_{S}|_{M} → Ω_{S}|_{M} identifies T_{M} ⊂ T_{S}|_{M} with T_{M}^{⊥} ⊂ Ω_{S}|_{M}. An equivalent condition is that the restriction of σ to a 2-form on M vanishes and that dim M = n. More generally, we define an immersed Lagrangian, to be an unramified morphism i : M → S, where M is a manifold of dimension n, such that i^{∗}σ ∈ Ω_{M}^{2} vanishes.

Holomorphic coordinates x_{1}, …, x_{n}, p_{1}, …, p_{n} on S are called Darboux coordinates, if

σ = ∑_{i=1}^{n} dp_{i} ∧ dx_{i} .

Let us introduce one further piece of notation. For a subbundle E ⊂ Ω_{S} we consider the associated bundles E^{⊥}, E^{∨} and E^{†} defined the short exact sequences of vector bundles

\xymatrix{0 \ar[r] & E^{⊥} \ar[r] & T_{S} \ar[r] & E^{∨} \ar[r] & 0},

and

\xymatrix{0 \ar[r] & E \ar[r] & Ω_{S} \ar[r] & E^{†} \ar[r] & 0}.

2.1 Lagrangian foliations

Definition 2.1. A Lagrangian foliation on S is an integrable distribution F ⊂ T_{S}, where F ⊂ T_{S} is a Lagrangian subbundle, i.e., X → X ⌟ σ defines an isomorphism of vector bundles F → F^{⊥} ⊂ Ω_{S}.

All leaves of the Lagrangian foliation F are Lagrangian submanifolds of S. The Lagrangian foliation F ⊂ T_{S} may be equivalently defined in terms of the subbundle E = F^{⊥} ⊂ Ω_{S}. Usually, we find it more convenient to specify E ⊂ Ω_{S}, rather than F ⊂ T_{S}. In terms of E, we have the following isomorphism of short exact sequences of vector bundles:

\xymatrix{0 \ar[r] & E^{⊥} \ar[d]_{≅} \ar[r] & T_{S} \ar[r] \ar[d]^{⌟ σ}_{≅} & E^{∨} \ar[r] \ar[d]_{≅} & 0 \\ 0 \ar[r] & E \ar[r] & Ω_{S} \ar[r] & E^{†} \ar[r] & 0}

Definition 2.2. . A polarized symplectic manifold is a symplectic manifold endowed with a Lagrangian foliation.

The canonical partial connection

Any foliation F ⊂ T_{S} defines a partial connection on the quotient bundle T_{S}/F:

∇ : T_{S}/F ⟶ F^{∨} ⊗ T_{S}/F ,  (11)

given by

∇_{Y} (X) = [Y, X] ,

for Y ∈ F and X ∈ T_{S}/F. This partial connection is flat. The dual bundle of T_{S}/F is F^{⊥} ⊂ Ω_{S}. The dual connection

∇ : F^{⊥} ⟶ F^{∨} ⊗ F^{⊥}

is given by

∇_{Y} (ω) = Y ⌟ dω

for Y ∈ F and ω ∈ F^{⊥} ⊂ Ω_{S}.

Let us specialize to the case that F is Lagrangian. Then we can transport the partial connection from F^{⊥} to F, via the isomorphism F ≅ F^{⊥}. We obtain the canonical partial flat connection

∇ : F ⟶ F^{∨} ⊗ F

characterized by

∇_{Y} (X) ⌟ σ = Y ⌟ d(X ⌟ σ) ,

for Y ∈ F and X ∈ F. The dual of this partial connection is

∇ : F^{∨} ⟶ F^{∨} ⊗ F^{∨} ,

which is characterized by

∇_{Y} (X ⌟ σ) = [Y, X] ⌟ σ ,

for Y ∈ F and X ∈ T_{S}.

Oriented Lagrangian foliations

Definition 2.3. Let F ⊂ T_{S} a Lagrangian foliation on S. An orientation of F is a nowhere vanishing global section

θ ∈ Γ (S, Λ^{n}F) ,

which is flat with respect to the canonical partial connection on F.

A polarized symplectic manifold is called oriented, if its Lagrangian foliation is endowed with an orientation.

Remark 2.4. If θ is an orientation of the Lagrangian foliation F, then we have ∇(θ ⌟ σ^{n}) = 0. (Note that θ ⌟ σ^{n} ∈ Λ^{n}F^{⊥} ⊂ Λ^{n}Ω_{S}.)

2.2 Polarizations and transverse Lagrangians

Let E ⊂ Ω_{S} define a Lagrangian foliation on S.

Lemma 2.5. Let M be a Lagrangian submanifold of S which is everywhere transverse to E. Then there exists (locally near M) a unique section s of E, such that ds = σ and M = Z(s), i.e., M is the zero locus of s (as a section of the vector bundle E).

Definition 2.6. We call s the Euler form of M with respect to E, or the Euler section of M in E.

Remark 2.7. Conversely, if s is any section of E such that ds = σ, then Z(s) is a Lagrangian submanifold. Thus we have a canonical one-to-one correspondence between sections s of E such that ds = σ and Lagrangian submanifolds of S transverse to E.

Lemma 2.8. Let (S, F, σ, θ) be an oriented polarized symplectic manifold and E = F^{⊥}. Let M ⊂ S be a Lagrangian submanifold, everywhere transverse to F. Then near every point of M there exists a set of Darboux coordinates x_{1}, …, x_{n}, p_{n} …, p_{n} such that

(i) M = Z(p_{1}, …, p_{n}), (ii) F = ⟨\frac{∂}{∂p_{1}} …, \frac{∂}{∂p_{n}} ⟩

(iii) ∇(\frac{∂}{∂p_{i}}) = 0, for all i = 1, …, n, (iv) θ = \frac{∂}{∂p_{i}} ∧ … ∧ \frac{∂}{∂p_{n}}, Moreover, in these coordinates we have

(i) E = ⟨dx_{1}, …, dx_{n}⟩, (ii) the Euler form s of M inside E is given by s = ∑ p_{i}dx_{i}.

3 Derived Lagrangian intersections on polarized symplectic manifolds

Definition 3.1. Let (S, E, σ) be a polarized symplectic manifold and L, M immersed Lagrangians of S which are both transverse to E. Then the derived intersection

L ⋒_{S} M

is the sheaf of differential Gerstenhaber algebras (ΛT_{M}, t̃) on M, where t̃ is the derivation on ΛT_{M} induced by the restriction to M of the Euler section t ∈ E ⊂ Ω_{S} of L.

Since dt = σ, and M is Lagrangian, the restriction of t to M is closed, and so t̃ is a derivation with respect to the Schouten-Nijenhuis bracket on ΛT_{M}, making (ΛT_{M}, t̃) a differential Gerstenhaber algebra.

Remark 3.2. After passing (locally in L) to suitable étale neighborhoods of L in S we can assume that L is embedded (not just immersed) in S and that L admits a globally defined Euler section t on S. This defines the derived intersection étale locally in M, and the global derived intersection is defined by gluing in the étale topology on M.

Remark 3.3. If we forget about the bracket, the underlying complex of O_{S}- modules (ΛT_{M}, t̃) represents the derived tensor product

O_{L} ⊗^{L}_{O_{S}} O_{M}

in the derived category of sheaves of O_{S}-modules.

Remark 3.4. The derived intersection L ⋒_{S} M depends a priori on the polarization E. We will see later (see the proof of Theorem 4.2) that different polarizations lead to locally quasi-isomorphic derived intersections. (The quasi-isomorphism is not canonical, as it depends on the choice of a third polarization transverse to both of the polarizations being compared. It is not clear that such a third polarization can necessarily be found globally.)

Remark 3.5. The derived intersection does not seem to be symmetric. We will see below that L ⋒_{S} M = M ⋒\overline{S} L, where \overline{S} = (S, −σ), but only if \overline{S} is endowed with a different polarization, transverse to E. Then the issue of change of polarization of Remark 3.4 arises.

Definition 3.6. Let S, L, M be as in Definition 3.1. Let M be oriented, i.e., endowed with a nowhere vanishing top-degree differential form ω_{M}^{∘}. (Since M is Lagrangian, this amounts to the same as a trivialization of the determinant of the normal bundle N_{M/C}.) We call the differential Batalin-Vilkovisky algebra (ΛT_{M}, t̃, [ ], d^{∘}), where d^{∘} is induced by ω_{M}^{∘} as in Section 1.5, the oriented derived intersection, notation L ⋒_{S}^{∘} M.

By a local system we mean a vector bundle (locally free sheaf of finite rank) endowed with a flat connection. Every local system P on a complex manifold M has an associated holomorphic de Rham complex (P ⊗_{O_{M}} Ω_{M}^{•}, d), where d denotes the covariant derivative.

Definition 3.7. Let (S, E, σ) be a polarized symplectic manifold and L, M immersed Lagrangians, both transverse to E. Let P be a local system on M and Q a local system on S. The derived hom from Q|L to P |M is the differential Batalin-Vilkovisky module

ℝHom_{S} (Q|L, P |M) = (Ω_{M}^{•} ⊗ Q^{∨}|_{M} ⊗ P, t)

over the differential Gerstenhaber algebra

L ⋒_{S} M = (ΛT_{M}, t̃) .

The tensor products are taken over O_{M}. The closed 1-form t ∈ Ω_{M} is the restriction to M of the Euler section of L inside E. The O_{M}-linear differential t is multiplication by t and the ℂ-linear differential d is covariant derivative with respect to the induced flat connection on Q^{∨}|_{M} ⊗ P.

Remark 3.8. If we forget about the ℂ-linear differential d and the flat connections on P and Q, the underlying complex of O_{S}-modules ℝHom_{S}(Q|L, P |M) represents the derived sheaf of homomorphisms ℝHom_{O_{S}}(Q|_{L}, P) in the derived category of sheaves of O_{S}-modules.

3.1 The exchange property: Gerstenhaber case

Given two symplectic manifolds S′, S, of dimensions 2n′ and 2n, a symplectic correspondence between S′ and S is a manifold C of dimension n + n′, together with morphisms π′ : C → S′ and π : C → S, such that

(i) π^{∗}σ = π′^{∗}σ′ (as sections of Ω_{C}), (ii) C → S′ × S is unramified. Thus a symplectic correspondence is an immersed Lagrangian of

\overline{S}′ × S = (S′ × S, σ − σ′) .

Let C → S′×S be a symplectic correspondence. We say that the immersed Lagrangian L → S is transverse to C, if (i) for every (Q, P) ∈ C ×_{S} L we have that

T_{C}|_{Q} ⊕ T_{L}|_{P} ⟶ T_{S}|_{π(Q)}

is surjective, hence the pullback L′ = C ×_{S} L is a manifold of dimension n′

(ii) the natural map L′ → S′ is unramified (and hence L′ is an immersed

Lagrangian of S′). By exchanging the roles of S and S′ we also get the notion of transversality to C for immersed Lagrangians of S′.

Exchange property setup

Let (S, E, σ) and (S′, E′, σ′) be polarized symplectic manifolds. Let E^{⊥} ⊂ T_{S} and E′^{⊥} ⊂ T_{S′} be the corresponding Lagrangian foliations. Consider a transverse symplectic correspondence C → S′ × S. This means that C → S′ × S is transverse to the foliation E′^{⊥} × E^{⊥} of S′ × S. In particular, the composition

T_{C} ⟶ π^{∗}T_{S} ⟶ π^{∗}E^{∨}

is surjective. Hence the foliation E^{⊥} ⊂ T_{S} pulls back to a foliation F ⊂ T_{C} of rank n′. We have the exact sequence of vector bundles

0 ⟶ F ⟶ T_{C} ⟶ π^{∗}E^{∨} ⟶ 0 .  (12)

Similarly, the foliation E′^{⊥} ⊂ T_{S′} pulls back to a foliation F′ ⊂ T_{C} of rank n with the exact sequence

0 ⟶ F′ ⟶ T_{C} ⟶ π′^{∗}E′^{∨} ⟶ 0 .

Moreover, F and F′ are transverse foliations of C and so we have

F ⊕ F′ = T_{C} = π′^{∗}E′^{∨} ⊕ π^{∗}E^{∨} .

Even though it is not strictly necessary, we will make the assumption that F ⊂ T_{C} descends to a Lagrangian foliation F̃ ⊂ T_{S′} and F′ ⊂ T_{C} descends to a Lagrangian foliation F̃′ ⊂ T_{S}. This makes some of the arguments simpler.

Remark 3.9. The composition

\xymatrix{F \ar[r] & T_{C} \ar[r] & π′^{∗}T_{S′} \ar[r]^{⌟ π′^{∗}σ′} & π′^{∗}Ω_{S′} \ar[r] & π′^{∗}E′^{†}}

defines an isomorphism of vector bundles β : F \underrightarrow{∼} π′^{∗}E′^{†}, and its inverse η : π′^{∗}E′^{†} \underrightarrow{∼} F. We can reinterpret these as perfect pairings β : F ⊗_{O_{C}} π′^{∗}E′^{⊥} → O_{C} and η : F^{∨} ⊗_{O_{C}} π′^{∗}E′^{†} → O_{C}. These will be important in the proof below.

Now assume given immersed Lagrangians L of S and M′ of S′. Assume both are transverse to C. Then we obtain manifolds L′ and M by the pullback diagram

[figure: L′; □; L; M; □; C; π; π′; S; M′; S′]  (13)

Then L′ is an immersed Lagrangian of S′ and M an immersed Lagrangian of S.

Finally, we assume that L and M are transverse to E and that M′ and L′ are transverse to E′. As a consequence, L′ is transverse to F′ and M is transverse to F.

Remark 3.10. As M is transverse to F, we have a canonical isomorphism F |_{M} = N_{M/C}. Also, since π′^{∗}N_{M′/S′} = N_{M/C}, we have π′^{∗}E′^{⊥}|_{M} = N_{M/C}. Thus, restricting the pairings β and η to M, we obtain: β|_{M} : N_{M/C} ⊗_{O_{M}} N_{M/C} → O_{M} and η|_{M} : N_{M/C}^{∨} ⊗_{O_{M}} N_{M/C}^{∨} → O_{M}.

Lemma 3.11. If s ∈ E is the Euler section of M in E and s′ the Euler section of M′ in E′, then the homomorphism β|_{M} : N_{M/C} → N_{M/C}^{∨} fits into the commutative diagram

\xymatrix{T_{C} \ar[rr]^{s̃−s′} \ar[d] &  & O_{C} \ar[r]^{d} & Ω_{C} \ar[d] \\ T_{C}|_{M} \ar[r] & N_{M/C} \ar[r]^{β|_{M}} & N_{M/C}^{∨} \ar[r] & Ω_{C}|_{M}}

Proof. Let P ∈ M ⊂ C be a point. It suffices to prove the claim locally near P. Let F̃ ⊂ T_{S′} be the Lagrangian foliation on S′, which pulls back to F ⊂ T_{C}. Then F̃ is transverse to both E′ and M′.

Choose holomorphic functions x_{1}, …, x_{n′} in a neighborhood of π′(P) in S′, such that dx_{1}, …, dx_{n′} is a basis for E′ ⊂ Ω_{S′}. Also, choose y_{1}, …, y_{n′}, such that dy_{1}, …, dy_{n′} is a basis for F̃^{⊥} ⊂ Ω_{S′}. Then (x_{i}, y_{j}) is a set of coordinates for S′ near π′(P).

Let \overline{s} be the Euler section of M′ in F̃^{⊥} and f be the unique holomorphic function on S′, defined in a neighborhood of π′(P), such that f(π′(P)) = 0 and df = \overline{s}−s′. Then we have \overline{s} = ∑_{j} \frac{∂f}{∂y_{j}} dy_{j} and s′ = − ∑_{i} \frac{∂f}{∂x_{i}} dx_{i}. Moreover,

σ′ = ∑_{i,j} \frac{∂^{2}f}{∂x_{i}∂y_{j}} dx_{i} ∧ dy_{j}.

We remark that the composition T_{S′} →^{df} O_{S′} →^{d} Ω_{S′} factors through N_{M′/S′} → N_{M′/S′}^{∨}, because f vanishes on M′. The resulting map is, in fact, the Hessian of f. Via our identifications, this Hessian agrees with the map F̃|_{M′} → E′^{†}|_{M′} induced by σ′, because, with our choice of coordinates, F̃|_{M′} = N_{M′/S′} has basis \frac{∂}{∂x_{i}} and E′^{†}|_{M′} = N_{M′/S′}^{∨} has basis dy_{j}.

To transfer this result from S′ to C, we remark that the pullback of \overline{s} to C is necessary equal to the pullback of s to C. Thus the composition d ∘ (s̃− s′) is equal to the Hessian of the pullback of f to C. This is, by what we proved above, equal to the pullback of the map induced by σ′. ⊓ ⊔

Theorem 3.12. There are canonical quasi-isomorphisms of differential Gerstenhaber algebras

(M′ × L) ⋒_{\overline{S}′×S} C ⟶ L ⋒_{S} M ,

and

(M′ × L) ⋒_{\overline{S}′×S} C ⟶ M′ ⋒\overline{S}′ L′ .

In particular, the derived intersections L ⋒_{S} M and M′ ⋒\overline{S}′ L′ are canonically quasi-isomorphic.

Proof. Passing to étale neighborhoods of L in S and M′ in S′ will not change anything about either derived intersection L ⋒_{S} M or M′ ⋒\overline{S}′ L′, so we may assume, without loss of generality, that

(i) L is embedded (not just immersed) in S (and same for M′ in S′), (ii) L admits a global Euler section t with respect to E on S (and M′ has the

Euler section s′ in E′ on S′)

Then the Euler section of M′ with respect to E′ on \overline{S}′ is −s′. Thus the derived intersection L⋒_{S} M is equal to (ΛT_{M}, t̃) and the derived intersection M′⋒\overline{S}′ L′ equals (ΛT_{L′}, −s′).

Pulling back the 1-form t via π, we obtain a 1-form on C, which we shall, by abuse of notation, also denote by t. Similarly, pulling back s′ via π′ we get the 1-form s′ on C. The difference t − s′ is closed on C, and thus we have the differential Gerstenhaber algebra (ΛT_{C}, t̃− s′). We remark that it is equal to (M′ × L) ⋒_{\overline{S}′×S} C.

Recall that we have the identification T_{C} = π′^{∗}E′^{∨} ⊕ π^{∗}E^{∨}. Under this direct sum decomposition t̃− s′ splits up into two components, −s′ and t̃. Hence we obtain the decomposition

(ΛT_{C}, t̃− s′) = π′^{∗}(ΛE′^{∨}, −s′) ⊗ π^{∗}(ΛE^{∨}, t̃)

of differential graded O_{C}-algebras.

Recall that O_{S′} → O_{M′} induces a quasi-isomorphism of differential graded O_{S′}-algebras (ΛE′^{∨}, −s′) → O_{M′}. Because the pullback M = M′ ×_{S′} C is transverse, we get an induced quasi-isomorphism

π′^{∗}(ΛE′^{∨}, −s′) ⟶ O_{M}

of differential graded O_{C}-algebras. Tensoring with π^{∗}(ΛE^{∨}, t̃), we obtain the quasi-isomorphism

(ΛT_{C}, t̃− s′) ⟶ (ΛE^{∨}, t̃)|_{M} .

Noting that E^{∨}|_{M} = T_{M}, because M is an immersed submanifold in S transverse to E, we see that (ΛE^{∨}, t̃)|_{M} = (ΛT_{M}, t̃) and so we have a quasiisomorphism of differential graded O_{C}-algebras

ϕ : (ΛT_{C}, t̃− s′) ⟶ (ΛT_{M}, t̃) .  (14)

For analogous reasons, we also have the quasi-isomorphism

ϕ′ : (ΛT_{C}, t̃− s′) ⟶ (ΛT_{L′}, −s′) .

The proof will be finished, if we can enhance ϕ and ϕ′ by brackets, making them morphisms of differential Gerstenhaber algebras. We will concentrate on ϕ. The case of ϕ′ follows by symmetry.

Thus we shall define a bracket

{ } : ΛT_{C} ⊗_{ℂ} ΛT_{C} ⟶ ΛT_{M} ,  (15)

such that (ϕ, {, }) becomes a morphism of differential Gerstenhaber algebras. We use the foliation F ⊂ T_{C}. It defines (11) a partial flat connection

∇ : T_{C}/F ⟶ F^{∨} ⊗_{O_{C}} T_{C}/F .

By the usual formulas we can transport ∇ onto the exterior powers of T_{C}/F. In our context, we obtain

∇ : π^{∗}ΛE^{∨} ⟶ F^{∨} ⊗_{O_{C}} π^{∗}ΛE^{∨} .

To get the signs right, we will consider the elements of the factor F^{∨} in this expression to have degree zero.

Let us write the projection ΛT_{C} → π^{∗}ΛE^{∨} as ρ. We identify F^{∨}|_{M} with N_{M/C}^{∨} and (π^{∗}ΛE^{∨})|_{M} with ΛT_{M}. Then ϕ is the composition of ρ with restriction to M. We now define for X, Y ∈ ΛT_{C}

{X, Y } = η(∇(ρX)|_{M} ∧ ∇(ρY)|_{M}) .  (16)

In this formula, ‘∧’ denotes the homomorphism (all tensors are over O_{M})

(N_{M/C}^{∨} ⊗ ΛT_{M}) ⊗ (N_{M/C}^{∨} ⊗ ΛT_{M}) ⟶ (N_{M/C}^{∨} ⊗ N_{M/C}^{∨}) ⊗ ΛT_{M}
v ⊗ X ⊗ w ⊗ Y ⟼ v ⊗ w ⊗ X ∧ Y .

There is no sign correction in this definition, because the elements of N_{M/C}^{∨} are considered to have degree zero, by our sign convention. We have also extended the map η linearly to

η : (N_{M/C}^{∨} ⊗_{O_{M}} N_{M/C}^{∨}) ⊗_{O_{M}} ΛT_{M} ⟶ ΛT_{M} .

Claim. The conditions of Definition 1.12 are satisfied by (ϕ, { }).

All but the last condition follow easily from the definitions. Let us check Condition (iv). We use Remark 1.14. The ℂ-algebra ΛT_{C} is generated in degrees 0 and −1. As generators in degree −1, we may take the basic vector fields of a coordinate system for C. We choose this coordinate system such that M is cut out by a subset of the coordinates. Then, if we plug in generators of degree −1 for both X and Y in Formula (4), every term vanishes. Also, if we plug in terms of degree 0 for both X and Y, both sides of (4) vanish for degree reasons. By symmetry, we thus reduce to considering the case where X is of degree −1, i.e., a vector field on C, and Y is of degree 0, i.e., a regular function on C.

Hence we need to prove that for all X ∈ T_{C} and g ∈ O_{C} we have

X(g)|_{M} − ρ(X)|_{M}(g|_{M}) = {(t̃− s′)X, g} − t̃{X, g} .  (17)

Let s denote the Euler section of M in E ⊂ Ω_{S}, and its pullback to C. We will prove that

X(g)|_{M} − ρ(X)|_{M}(g|_{M}) = {(s̃− s′)X, g}  (18)

and

{(t̃− s̃)X, g} = t̃{X, g} .  (19)

Equation (18) involves only M, not L, and Equation (19) involves only E, not E′. Together, they imply Equation (17).

All terms in these three equations are O_{S}-linear in X and derivations in g, and may hence be considered as O_{C}-linear maps T_{C} → Der(O_{C}, O_{M}). As Der(O_{C}, O_{M}) = Hom_{O_{C}} (Ω_{C}, O_{M}) = T_{C}|M, we may also think of them as O_{C}-linear maps T_{C} → T_{C}|_{M}.

For example, the O_{C}-linear map

T_{C} ⟶ T_{C}|_{M}  (20)
X ⟼ {(s̃− s′)X, · }

Is equal to the composition

\xymatrix{T_{C} \ar[r]^{s̃−s′} & O_{C} \ar[r]^{d} & Ω_{C} \ar[r] & F^{∨}|_{M} \ar[r]^{η} & F |_{M} \ar[r] & T_{C}|_{M}} .

Then the commutative diagram (Lemma 3.11)

\xymatrix{T_{C} \ar[r]^{s̃−s′} \ar[d] & O_{C} \ar[r]^{d} & Ω_{C} \ar[d] \ar[dr] \\ T_{C}|_{M} \ar[d] &  & Ω_{C}|_{M} \ar[r] & F^{∨}|_{M} \ar[r]^{η} & F |_{M} \ar[r] & T_{C}|_{M} \\ N_{M/C} \ar[rr]^{β} &  & N_{M/C}^{∨} \ar[u] \ar[ur]_{id}}

and the fact that η is the inverse of β, proves that (20) is equal to the composition

\xymatrix{T_{C} \ar[r] & T_{C}|_{M} \ar[r]^{p} & T_{C}|_{M}} ,

where p is the projection onto the the second summand of the decomposition

T_{C}|_{M} = T_{M} ⊕ N_{M/C}

given by the foliation F transverse to M in C. If we denote by q the projection onto the first summand, we see that the map

T_{C} ⟶ T_{C}|_{M}  (21)
X ⟼ ρ(X)|_{M}( · |_{M})

is equal to

\xymatrix{T_{C} \ar[r] & T_{C}|_{M} \ar[r]^{q} & T_{C}|_{M}} .

Thus (20) and (21) sum up to the restriction map T_{C} → T_{C}|_{M}, which is equal to the map given by X ↦ X( ·)|_{M}. This proves (18).

Now, let us remark that for any closed 1-form u on C we have

ũ[Y, X] = Y (ũ(X)) − X(ũ(Y)) .

If u ∈ π^{∗}E ⊂ Ω_{C}, then ũ(Y) = 0, for all Y ∈ F. So if Y ∈ F we have

ũ[Y, X] = Y (ũ(X)) .

We have ũ[Y, X] = ũ(∇(X)(Y)) by definition of the partial connection ∇ and we can write Y (ũ(X)) = ⟨Y, d(ũ(X))⟩. In other words, the diagram

\xymatrix{T_{C}/F \ar[rr]^{∇} \ar[d]_{ũ} &  & Hom(F, T_{C}/F) \ar[d]^{∘ũ} \\ O_{C} \ar[r]^{d} & Ω_{C} \ar[r] & F^{∨}}

commutes. Thus, the larger diagram

\xymatrix{T_{C} \ar[r]^{ρ} \ar[dr]_{ũ} & π^{∗}E^{∨} \ar[r]^{∇} \ar[d]^{ũ} & F^{∨} ⊗ π^{∗}E^{∨} \ar[d]^{id ⊗ũ} \ar[r] & F^{∨}|_{M} ⊗ T_{M} \ar[r]^{η⊗id} \ar[d]^{id ⊗ũ|_{M}} & F |_{M} ⊗ T_{M} \ar[d]^{id ⊗ũ|_{M}} \\  & O_{C} \ar[r]^{d} & F^{∨} \ar[r] & F^{∨}|_{M} \ar[r]^{η} & F |_{M} \ar[d] \\  &  &  &  & T_{C}|_{M}}

commutes as well. We can apply these considerations to u = t − s. Then ũ = t̃ − s̃ and ũ|_{M} = t̃|_{M}. Thus the upper composition in this diagram represents the right hand side of Equation (19), and the lower composition represents the left hand side of Equation (19). This exhibits that (19) holds and finishes the proof of the theorem. ⊓ ⊔ 3.2 The Batalin-Vilkovisky case

For the exchange property in the Batalin-Vilkovisky case, we require an orientation on the symplectic correspondence C → S′ × S.

Definition 3.13. Let π : C → S be a morphism of complex manifolds, F ⊂ T_{C} and F̃ ⊂ T_{S} foliations. The foliations F, F̃ are compatible (with respect to π), if F → T_{C} → π^{∗}T_{S} factors through π^{∗}F̃ → π^{∗}T_{S}.

If F and F̃ are compatible, then partial connections with respect to F̃ pull back to partial connections with respect to F.

Now let (S, E, σ), (S′, E′, σ′) and C → S′ × S be, as in Section 3.1, polarized symplectic manifolds with a transverse symplectic correspondence. Let F and F′ be, as in 3.1, the inverse image foliations:

[figure: F; □; E^{⊥}; F′; □; C; π; π′; S; _{E′_{⊥}}; S′]  (22)

Furthermore, we suppose that F̃ ⊂ T_{S′} is a Lagrangian foliation on S′ compatible with F via π′ and that F̃′ ⊂ T_{S} is a Lagrangian foliation on S, compatible with F′ via π. Since the composition F → π′^{∗}T_{S′} → π′^{∗}E′^{∨} is an isomorphism, the map F → π′^{∗}T_{S′} identifies F with a subbundle of π′^{∗}T_{S′}. As F and F̃ have the same rank, it follows that F → π′^{∗}F̃ is an isomorphism of subbundles of π′^{∗}T_{S′}. Similarly, we have and identification F′ → π^{∗}F̃′ of subbundles of π^{∗}T_{S}.

Thus we have two Lagrangian foliations on \overline{S}′ × S, namely E′^{⊥} × E^{⊥} and F̃×F̃′. Both Lagrangian foliations are transverse to C, and they are transverse to each other near C.

Definition 3.14. If θ ∈ Γ (S′, Λ^{n′} F̃) and θ′ ∈ Γ (S, Λ^{n}F̃′) are orientations of the Lagrangian foliations F̃ on S′ and F̃′ on S, we call the data (F̃, θ, F̃′, θ′) an orientation of the symplectic correspondence C → S′ × S.

We call the transverse symplectic correspondence of polarized symplectic manifolds C → S′ × S orientable, if it admits an orientation.

Exchange property setup

Let (S, E, σ) and (S′, E′, σ′) be polarized symplectic manifolds and C → S′ × S, (F̃, θ, F̃′, θ′) an oriented transverse symplectic correspondence. Moreover, let L → S and M′ → S′ be, as in Section 3.1, immersed Lagrangians transverse to C, such that the induced M and L′ are transverse to E and E′, respectively. (This latter condition is satisfied if M′ and L are transverse to F̃ and F̃′, respectively.) Pulling back θ to C gives us a trivialization of Λ^{n′}F and restricting further to M gives a trivialization of the determinant of the normal bundle Λ^{n′}N_{M/C}, because of the canonical identification F |_{M} = N_{M/C}. Similarly, θ′ gives rise to a trivialization of the determinant of the normal bundle Λ^{n}N_{L′/C}.

Finally, let P′ be a local system on S′ and Q a local system on S. Let P = π′^{∗}P′ and Q′ = π^{∗}Q be the pullbacks of these local systems to C.

Theorem 3.15. There exists a canonical quasi-isomorphism of differential Batalin-Vilkovisky modules

ℝHom_{\overline{S}′×S} (O|(M′ × L), (P ⊗ Q′^{∨})|C) ⟶ ℝHom_{S} (Q|L, P |M)

of degree −n′, covering the corresponding canonical quasi-isomorphism of differential Gerstenhaber algebras of Theorem 3.12. Moreover, there is the quasiisomorphism of differential Batalin-Vilkovisky modules

ℝHom_{\overline{S}′×S} (O|(M′ × L), (P ⊗ Q′^{∨})|C) ⟶ ℝHom\overline{S}′ (P′^{∨}|M′, Q′^{∨}|L′)

of degree −n, covering the other canonical quasi-isomorphism of differential Gerstenhaber algebras of Theorem 3.12.

Thus, the derived homs ℝHom_{S}(Q|L, P |M) and ℝHom\overline{S}′(P′^{∨}|M′, Q′^{∨}|L′) are canonically quasi-isomorphic, up to a degree shift n′ − n.

Proof. Let t and s′ be as in the proof of Theorem 3.12. We need to construct quasi-isomorphisms of Batalin-Vilkovisky modules

(ψ, δ) : (Ω_{C}^{•} ⊗ P ⊗ Q′^{∨}, t − s′) ⟶ (Ω_{M}^{•} ⊗ P ⊗ Q′^{∨}, t)

and

(ψ′, δ′) : (Ω_{C}^{•} ⊗ P ⊗ Q′^{∨}, t − s′) ⟶ (Ω_{L′}^{•} ⊗ P ⊗ Q′^{∨}, −s′) .

(Note that because the elements of P and Q′^{∨} have degree zero, it is immaterial in which order we write the two factors P and Q′^{∨}.) The case of (ψ′, δ′) being analogous, we will discuss only (ψ, δ).

Let us start with ψ. Denote the pullback of the orientation θ ∈ Λ^{n′} F̃ to C by the same letter, thus giving us a trivialization θ ∈ Λ^{n′}F. Note that contracting α ∈ Ω_{C}^{•} with θ gives a form θ ⌟ α in the subbundle π^{∗}ΛE ⊂ Ω_{C}^{•}, see (12). Recall the non-degenerate symmetric bilinear form β : N_{M/C} ⊗_{O_{M}} N_{M/C} → O_{M} of Remark 3.10. Since F |_{M} = N_{M/C}, we may apply the discriminant of β to θ|_{M} ⊗ θ|_{M} to obtain the nowhere vanishing regular function

g = det β (θ|_{M} ⊗ θ|_{M}) ∈ O_{M}  (23)

on M. The homomorphism ψ is now defined as the composition

\xymatrix{ψ : Ω_{C}^{•} \ar[r]^{θ ⌟ ·} & π^{∗}ΛE \ar[r]^{res |_{M}} & ΛE|_{M} = Ω_{M}^{•} \ar[r]^{·g} & Ω_{M}^{•}} .

Tensoring with P ⊗Q′^{∨}, we obtain the quasi-isomorphism of differential graded modules

ψ : (Ω_{C}^{•} ⊗ P ⊗ Q′^{∨}, t − s) ⟶ (Ω_{M}^{•} ⊗ P ⊗ Q′^{∨}, t)

covering the morphism of differential graded algebras ϕ of (14). The formula for ψ is

ψ(α) = g · ⌈θ ⌟ α⌉_{M} .  (24)

(‘Ceiling brackets’ denote restriction.) Note that deg ψ = −n′

Let us next construct δ : Ω_{C}^{•} → Ω_{M}^{•}. Recall the canonical partial flat connection on the Lagrangian foliation F̃ on S′:

˜∇ : F̃ ⟶ F̃^{∨} ⊗_{O_{S′}} F̃ ,

defined by the requirement

˜∇Ỹ (X̃) ⌟ σ′ = Ỹ ⌟ d(X̃ ⌟ σ′) ,

for Ỹ, X̃ ∈ F̃. As F is compatible with F̃ via π′, we get the pullback partial flat connection

∇ : π′^{∗}F̃ ⟶ F^{∨} ⊗_{O_{C}} π′^{∗}F̃ .

Making the identification F = π′^{∗}F̃ we rewrite this partial connection as

∇ : F ⟶ F^{∨} ⊗_{O_{C}} F .

It is characterized by the formula

∇_{Y} (X) ⌟ σ = Y ⌟ d(X ⌟ σ) ,

for Y, X ∈ F. We have written σ for the restriction of the symplectic form σ′ to C. The dual connection

∇ : F^{∨} ⟶ F^{∨} ⊗_{O_{C}} F^{∨}  (25)

satisfies

∇_{Y} (X′ ⌟ σ) = [Y, X′] ⌟ σ ,

for Y ∈ F and X′ ∈ T_{C}.

Recall that we also have the partial connection

∇ : π^{∗}E ⟶ F^{∨} ⊗_{O_{C}} π^{∗}E  (26)

defined by ∇_{Y} ω = Y ⌟ dω, for Y ∈ F and ω ∈ π^{∗}E ⊂ Ω_{C}. We used the dual of this connection in the proof of Theorem 3.12.

Thus, we have partial flat connections on F and π^{∗}E, in such a way that the canonical homomorphism F → π^{∗}E given by X ↦ X ⌟ σ is flat. We hope there will be no confusion from using the same symbol ∇ for both partial connections. As usual, we get induced partial connections on all tensor operations involving F and π^{∗}E. We define

∇^{2} : π^{∗}ΛE ⟶ F^{∨} ⊗ F^{∨} ⊗ π^{∗}ΛE

as the composition (all tensor products are over O_{C})

π^{∗}ΛE ⟶^{∇} F^{∨} ⊗ π^{∗}ΛE ⟶^{∇} F^{∨} ⊗ F^{∨} ⊗ π^{∗}ΛE .

We will also need

∇^{3} : O_{C} ⟶ F^{∨} ⊗ F^{∨} ⊗ F^{∨} .

To simplify notation, let us assume that the closed 1-form s − s′ on C is exact. Let I be the ideal of M in O_{C}. Then there exists a unique regular function f ∈ I^{2}, such that df = s − s′. The fact that f ∈ I^{2} follows because s and s′ vanish in Ω_{C}|_{M}, so df vanishes in Ω_{C}|_{M}. Then the Hessian of f is a symmetric bilinear form N_{M/C} ⊗_{O_{M}} N_{M/C} → O_{M}, and is equal to β|_{M}, by Lemma 3.11..

Finally, we define δ : Ω_{C}^{•} → Ω_{M}^{•} as a certain ℂ-linear combination of the two compositions

\xymatrix{Ω_{C}^{•} \ar[r]^{θ ⌟ ·} & π^{∗}ΛE \ar[r]^{∇^{2}} & F^{∨} ⊗_{O_{C}} F^{∨} ⊗_{O_{C}} π^{∗}ΛE \ar[d]^{res |_{M}} \\  &  & N_{M/C}^{∨} ⊗_{O_{M}} N_{M/C}^{∨} ⊗_{O_{M}} Ω_{M}^{•} \ar[r]^{η} & Ω_{M}^{•} \ar[r]^{· g} & Ω_{M}^{•} ,}

and

\xymatrix{π^{∗}ΛE \ar[r]^{∇} & F^{∨} ⊗ π^{∗}ΛE \ar[r]^{∇^{3}(f)⊗id} & (F^{∨})^{⊗4} ⊗ π^{∗}ΛE \ar[d]^{res |_{M}} \\ Ω_{C}^{•} \ar[u]_{θ ⌟ ·} &  & (N^{∨})^{⊗4} ⊗ Ω_{M}^{•} \ar[r]^{η⊗η} & Ω_{M}^{•} \ar[r]^{· g} & Ω_{M}^{•} .}

Here η and η ⊗ η are the linear extension of the map η from Remark 3.10. In fact, we define:

δ(α) = −\frac{1}{2}g · η(⌈∇^{2}(θ ⌟ α)⌉_{M} ) + \frac{1}{2}g · (η ⊗ η)(⌈∇^{3}(f) ⊗ ∇(θ ⌟ α)⌉_{M} ) .

As P and Q′^{∨} have flat connections on them, their pullbacks to C do, too. In particular, we can partially differentiate. Thus, δ extends naturally to the map δ : Ω_{C}^{•} ⊗ P ⊗ Q′^{∨} → Ω_{M} ⊗ P ⊗ Q′^{∨}.

We need to check Properties (i) and (ii) of Definition 1.24. To simplify notation, we will only spell out the case where P = Q′ = (O_{C}, d), leaving the general case to the reader.

Proving (ii) is a straightforward but tedious calculation using the properties of the partial connections on π^{∗}ΛE, π^{∗}ΛE^{∨}, F and F^{∨}, in particular, compatibility with contraction. One can simplify this calculation by using Remark 1.27: choose ℂ-algebra generators for ΛT_{C} in such a way that the degree −1 generators are flat for the partial connection (see below). This reduces to checking (8) for the case where X and Y are of degree 0, i.e., regular functions x and y on C. The claim is:

δ(xyω) + xyδ(ω) + {x, y}ψ(ω) = xδ(yω) + yδ(xω) ,

for all ω ∈ Ω_{C}. We leave the details to the reader, and only write down the terms containing dx ⊗ dy and only after canceling g · (θ ⌟ ω)|_{M}. In fact, from the term δ(xyω) we get the contribution

−\frac{1}{2}η⌈dx ⊗ dy + dy ⊗ dx⌉_{M} ,

and from the term {x, y}ψ(ω) we get the contribution

η⌈dx ⊗ dy⌉_{M} ,

and these two expressions do, indeed, add up to 0, because η is symmetric.

To prove (i), we shall use Remark 1.28. We will carefully choose a local trivialization of the vector bundle T_{C}, as this will give local generators for ΛT_{C} as O_{C}-algebra.

The equations we wish to prove can be checked locally. So we pick a point P ∈ M and pass to a sufficiently small analytic neighborhood of P in C.

Choose holomorphic functions p_{1}, …, p_{n′} in a neighborhood of π′(P) in S′ satisfying

(i) p_{1}, …, p_{n′} cut out the submanifold M′ ⊂ S′,

(ii) dp_{1}, …, dp_{n′} form a frame of F̃^{∨},

(iii) dp_{1}, …, dp_{n′} are flat for the partial connection on F̃^{∨}, (iv) θ ⌟ (dp_{1} ∧ … ∧ dp_{n′}) = 1. Denote the pullbacks of these functions to C by the same letters. Then these functions on C cut out M, their differentials are flat for the partial connection (25) and form a frame for F^{∨}. Also, the last property remains true as written. Such p_{i} exist by Lemma 2.8

Similarly, we choose holomorphic functions x_{1}, …, x_{n} in a neighborhood of π(P) in S, such that dx_{1}, …, dx_{n} form a frame for the subbundle E ⊂ Ω_{S}. Then the dx_{i} are automatically flat for the partial connection on E. Again, we denote the pullbacks to C of these functions by the same letters. For the functions x_{1}, …, x_{n} on C we have that their differentials dx_{1}, …, dx_{n} form a flat frame for the subbundle π^{∗}E of Ω_{C}. In particular, the restrictions of x_{1}, …, x_{n} to M ⊂ C form a set of coordinates for M near P.

Then the union of these two families dp_{1}, …, dp_{n′}, dx_{1}, …, dx_{n} forms a basis for Ω_{C}. We denote the dual basis (as usual) by \frac{∂}{∂p_{1}} , …, \frac{∂}{∂p_{n′}} , \frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}. We have

θ = (−1)^{\frac{1}{2} n′(n′−1)} \frac{∂}{∂p_{1}} ∧ … ∧ \frac{∂}{∂p_{n′}} .

We define

ω^{∘} = dp_{1} ∧ … ∧ dp_{n′} ∧ dx_{1} ∧ … ∧ dx_{n} ,

which is a basis for Ω_{C}^{•} as a ΛT_{C}-module. Note that

θ ⌟ ω^{∘} = dx_{1} ∧ … ∧ dx_{n} .

We denote the restriction to M of dx_{1} ∧ … ∧ dx_{n} by τ^{∘}. This is a basis for Ω_{M}^{•} as a ΛT_{M}-module. We have

⌈θ ⌟ ω^{∘}⌉_{M} = τ^{∘} .

We now have to prove that for X = \frac{∂}{∂p_{i}} and X = \frac{∂}{∂x_{j}} we have

ψ(d(X ⌟ ω^{∘})) − (−1)^{n′}d(ψ(X ⌟ ω^{∘}))

= −2(−1)^{n′}t ∧ δ(X ⌟ ω^{∘}) + 2 δ((t − s′) ∧ (X ⌟ ω^{∘})) . (27)

For any of our values for X we have d(X ⌟ ω^{∘}) = 0. So the first term in (27) always vanishes. Similarly, the third term always vanishes, because every of our values for X, as well as θ and ω^{∘} are flat for the partial connection ∇. Thus only the second and fourth term of (27) contribute.

Consider the fourth term. We have

δ((t − s′) ∧ (X ⌟ ω^{∘})) = δ((t − s) ∧ (X ⌟ ω^{∘})) + δ(df ∧ (X ⌟ ω^{∘}))
= δ((X ⌟ (t − s))ω^{∘})) + δ(X(f) ω^{∘}) .

Since t − s is a section of π^{∗}E and is also a closed 1-form, t − s is flat with respect to the partial connection ∇. Since X is by assumption also flat with respect to ∇, it follows that ∇(X ⌟ (t − s)) = 0, and hence the fourth term of (27) is equal to

δ((t − s′) ∧ (X ⌟ ω^{∘})) = δ(X(f) ω^{∘}) .

Thus, (27) reduces to

(−1)^{n′}d(ψ(X ⌟ ω^{∘})) = −2 δ(X(f) ω^{∘}) .  (28)

Let us first consider the case that X = \frac{∂}{∂p_{i}}. In this case θ ∧ X = 0, so that the left hand side of (28) vanishes. The claim is therefore, that

δ(\frac{∂f}{∂p_{i}} ω^{∘}) = 0 ,

for all i = 1, …, n′. This is equivalent to

η(⌈∇^{2} \frac{∂f}{∂p_{i}} ⌉_{M} ) = (η ⊗ η)(⌈∇^{3}f⌉_{M} ⊗ ⌈∇\frac{∂f}{∂p_{i}} ⌉_{M} ) .  (29)

To check (29), let us write it out in coordinates. The right hand side is equal to

∑_{k,l} ∑_{m,n} η_{kl}η_{mn} \frac{∂^{3}f}{∂p_{k}∂p_{l}∂p_{m}} |_{p=0} \frac{∂^{2}f}{∂p_{n}∂p_{i}} |_{p=0} = ∑_{k,l} ∑_{m} η_{kl}δ_{m}^{i} \frac{∂^{3}f}{∂p_{k}∂p_{l}∂p_{m}} |_{p=0}

= ∑_{k,l} η_{kl} \frac{∂^{3}f}{∂p_{k}∂p_{l}∂p_{i}} |_{p=0} ,

which is, indeed, equal to the left hand side of (29).

Now let us consider the case X = \frac{∂}{∂x_{j}}. Recall that the ideal I defining M

is given by I = (p_{1}, …, p_{n′}). Since f ∈ I^{2}, we still have \frac{∂f}{∂x_{j}} ∈ I^{2} and hence

∇(\frac{∂f}{∂x_{j}})|_{p=0} = 0. Thus, the right hand side of (28) is equal to

−2 δ(\frac{∂f}{∂x_{j}} ω^{∘}) = g · η(⌈∇^{2} \frac{∂f}{∂x_{j}} ⌉_{M} ) τ^{∘} + 0
= g · tr (η · \frac{∂}{∂x_{j}} H(f)) τ^{∘} ,

because differentiating with respect to x_{j} commutes with restriction to M = {p = 0}. (We have written H(f) for the Hessian of f.) On the other hand, the left hand side of (28) is equal to

(−1)^{n′}d(ψ(X ⌟ ω^{∘})) = \frac{∂g}{∂x_{j}} τ^{∘} ,

and thus, our final claim is equivalent to

\frac{∂g}{∂x_{j}} = g · tr (η · \frac{∂}{∂x_{i}} H(f)) .

Recalling that g = det H(f), and η is the inverse of H(f), this claim follows from the following:

Claim. Let A be an invertible square matrix of regular functions on the manifold M. Then for every vector field X on M we have

(det A)^{−1}X(det A) = tr (A^{−1}X(A)) .

This last claim is both well-known and easy to check. ⊓ ⊔

3.3 The oriented Batalin-Vilkovisky case

The setup is exactly the same as in Section 3.2, with one additional ingredient; namely an orientation of C, i.e., a nowhere vanishing global section ω_{C}^{∘} ∈ Ω_{C}^{n+n′}. We require ω_{C}^{∘} to be compatible with the orientation (F̃, F̃′, θ, θ′) on the symplectic correspondence C → S′ × S in the following sense: namely we ask that

(i) ∇(θ ⌟ ω_{C}^{∘}) = 0, where ∇ : π^{∗}E → F^{∨} ⊗ π^{∗}E is the partial connection (26) on F^{⊥} = π^{∗}E defined by the foliation F of C, (ii) ∇′(θ′ ⌟ ω_{C}^{∘}) = 0, where ∇′ : π′^{∗}E′ → F′^{∨} ⊗ π′^{∗}E′ is the corresponding

partial connection defined by the foliation F′ of C.

Now, (θ ⌟ ω_{C}^{∘})|_{M} is an orientation of M (recalling that (π^{∗}E)|_{M} = Ω_{M}). We shall denote it by ω_{M}^{∘}. Similarly, (θ′ ⌟ ω_{C}^{∘})|_{L′} is an orientation of L′, which we shall denote by ω_{L′}^{∘}. This orients the three Lagrangian intersections in Theorem 3.12.

Theorem 3.16. The quasi-isomorphisms of differential Gerstenhaber algebras of Theorem 3.12 are canonically enhanced to quasi-isomorphisms of differential Batalin-Vilkovisky algebras

(M′ × L) ⋒_{\overline{S}′×S}^{∘} C ⟶ L ⋒_{S}^{∘} M ,

and

(M′ × L) ⋒_{\overline{S}′×S}^{∘} C ⟶ M′ ⋒_{\overline{S}′}^{∘} L′ .

In particular, the oriented derived intersections L ⋒_{S}^{∘} M and M′ ⋒_{\overline{S}′}^{∘} L′ are canonically quasi-isomorphic.

Proof. In view of Theorems 3.12 and 3.15 and the results of Section 1.6, we only need to check that

(i) ψ(ω_{C}^{∘}) = ω_{M}^{∘}, (ii) ψ′(ω_{C}^{∘}) = ω_{L′}^{∘}, (iii) δ(ω_{C}^{∘}) = 0, (iv) δ′(ω_{C}^{∘}) = 0, where (ψ, δ) and (ψ′, δ′) are the homomorphisms of differential Batalin-Vilkovisky modules constructed in the proof of Theorem 3.15. But the first two follow from the above definitions and the last two from the above assumptions. ⊓ ⊔

Remark 3.17. If C = S and C → S × S is the diagonal, a canonical choice for the orientation of C is ω_{C}^{∘} = σ^{n}, by Remark 2.4. In this case, we also have ω_{M}^{∘} = θ|_{M}, via the identification Ω_{M}^{n} = Λ^{n}N_{M/S} = Λ^{n}F |_{M}. Similarly, ω_{L′}^{∘} = θ′|_{L′}.

4 The Gerstenhaber structure on Tor and the Batalin-Vilkovisky structure on Ext

4.1 The Gerstenhaber algebra structure on Tor

Let L and M be immersed Lagrangians in the symplectic manifold S. Write Tor_{O_{S}}^{i}(O_{L}, O_{M}) = Tor_{−i}^{O_{S}}(O_{L}, O_{M}). The direct sum

Tor_{O_{S}}^{•}(O_{L}, O_{M}) = ⨁_{i} Tor_{O_{S}}^{i}(O_{L}, O_{M})

is a graded sheaf of O_{S}-algebras, concentrated in non-positive degrees.

Remark 4.1. To be precise, we have to use the analytic étale topology on S to be able to think of Tor_{O_{S}}^{•}(O_{L}, O_{M}) as a sheaf of O_{S}-algebras. If L and M are embedded, not just immersed, we can use the usual analytic topology. Alternatively, introduce the fibered product Z = L ×_{S} M and think of Tor_{O_{S}}^{•}(O_{L}, O_{M}) as a sheaf of graded O_{Z}-algebras.

Theorem 4.2. There exists a unique bracket of degree +1 on Tor_{O_{S}}^{•}(O_{L}, O_{M}) such that

(i) Tor_{O_{S}}^{•}(O_{L}, O_{M}) is a sheaf of Gerstenhaber algebras, (ii) whenever E is a (local) polarization of S, such that L and M are transverse

to E, then this sheaf of Gerstenhaber algebras is obtained from the derived intersection L ⋒_{S} M (defined with respect to E) by passing to cohomology.

Proof. Without loss of generality, assume that L and M are submanifolds. For every point of S we can find an open neighborhood in S, over which we can choose a polarization E, which is transverse to L and M. This proves uniqueness.

For existence, we have to prove that any two polarizations E, E′′ give rise to the same bracket on Tor_{O_{S}}^{•}(O_{L}, O_{M}). This is a local question, so we may choose a third polarization E′, which is transverse to both E and E′′, and also to L and M.

We will apply the exchange property, Theorem 3.12, twice. First to the symplectic correspondence Δ : C = S → S × S between the polarized symplectic manifolds (S, E′) and (S, E), then to the symplectic correspondence Δ : C = \overline{S} → \overline{S} × \overline{S} between (\overline{S}, E′′) and (\overline{S}, E′). We obtain the following diagram of quasi-isomorphisms of sheaves of differential Gerstenhaber algebras:

\xymatrix{ & (M × L) ⋒_{\overline{S}×S}^{E′×E} S \ar[r] \ar[d] & L ⋒_{S}^{E} M \\ (L × M) ⋒_{S×\overline{S}}^{E′′×E′} \overline{S} \ar[r] \ar[d] & M ⋒E′ ; \overline{S} L \\ L ⋒_{S}^{E′′} M}  (30)

We have included the polarizations defining the derived intersections in the notation.

Passing to cohomology sheaves, we obtain the following diagram of isomorphisms of sheaves of Gerstenhaber algebras:

\xymatrix{ & Tor_{O_{S×S}}^{•}(O_{M×L}, O_{S}) \ar[r] \ar[d] & Tor_{O_{S}}^{•}(O_{L}, O_{M}) \\ Tor_{O_{S×S}}^{•}(O_{L×M}, O_{S}) \ar[r] \ar[d] & Tor_{O_{S}}^{•}(O_{M}, O_{L}) \\ Tor_{O_{S}}^{•}(O_{L}, O_{M})}

One checks that all these morphisms are the canonical ones, and hence that the composition of all four of them is the identity on Tor_{O_{S}}^{•}(O_{L}, O_{M}). If the identity preserves the two brackets on Tor_{O_{S}}^{•}(O_{L}, O_{M}) defined by E and E′′, respectively, then the two brackets are equal. ⊓ ⊔

4.2 The Batalin-Vilkovisky structure on Ext

Let L and M continue to denote immersed Lagrangians in the symplectic manifold S. Furthermore, let P be a local system on M and Q a local system on L. The direct sum

Ext_{O_{S}}^{•}(Q, P) = ⨁_{i} Ext_{O_{S}}^{i}(Q, P)

is a graded sheaf of Tor_{O_{S}}^{•}(O_{L}, O_{M})-modules.

Theorem 4.3. There exists a unique ℂ-linear differential

d : Ext_{O_{S}}^{i}(Q, P) ⟶ Ext_{O_{S}}^{i+1}(Q, P) ,

(for all i) such that

(i) Ext_{O_{S}}^{•}(Q, P) is a sheaf of Batalin-Vilkovisky modules over the Gerstenhaber algebra Tor_{O_{S}}^{•}(O_{L}, O_{M}),

(ii) whenever E is a (local) polarization of S, such that L and M are transverse to E, and \overline{Q} is a local system on S restricting to Q on L, this sheaf of Batalin-Vilkovisky modules is obtained from the derived hom ℝHom_{S}(\overline{Q}|L, P |M) (defined with respect to E) by passing to cohomology.

Proof. Uniqueness is clear. Let us prove existence. For this, we assume given two polarization E, E′′, transverse to L and M, and two extensions \overline{Q} and Q̂ of Q to S. To compare the derived homs ℝHom_{S}^{E}(\overline{Q}|L, P |M) and ℝHom_{S}^{E′′}(Q̂|L, P |M), we choose (locally) a third polarization E′, transverse to L and M, and E and E′′, and an extension \overline{P} of P to S. These choices make the five derived homs in Diagram (32), below, well-defined.

To define the homomorphisms of differential Batalin-Vilkovisky modules in (32), we orient the symplectic correspondence given by the diagonal of S in the canonical way, as in Remark 3.17, by σ^{n}. The corresponding symplectic correspondence given by the diagonal of \overline{S} is hence oriented by (−1)^{n}σ^{n}. We also orient the three Lagrangian foliations F, F′, F′′ on S, by choosing θ ∈ Λ^{n}F, θ′ ∈ Λ^{n}F′ and θ′′ ∈ Λ^{n}F′′. (Note that F̃ = F = E^{⊥}, etc., in our case.) But the choice of θ, θ′′ is not completely arbitrary. In fact, notice that both F |_{L} and F′′|_{L} are complements to T_{L} ⊂ T_{S}|_{L}, so that we get a canonical identification F |_{L} \underrightarrow{∼} F′′|_{L}. We choose θ and θ′′ in such a way that the composition

\xymatrix{O_{L} \ar[r]^{θ|_{L}} & det F |_{L} \ar[r]^{≅} & det F′′|_{L} \ar[r]^{θ′′^{∨}|_{L}} & O_{L}}  (31)

is equal to the identity.

Now, by applying the exchange property, Theorem 3.15, twice, as in the proof of Theorem 4.2, we obtain the following diagram of quasi-isomorphisms of differential Batalin-Vilkovisky modules, covering Diagram (30) of differential Gerstenhaber algebras:

\xymatrix{ℝHom_{\overline{S}×S}^{E′×E} (O|(M × L), (\overline{P} ⊗ \overline{Q}^{∨})|S) \ar[r] \ar@{-->}[dr] & ℝHom_{S}^{E}(\overline{Q}|L, P |M) \\ ℝHom_{S×\overline{S}}^{E′′×E′} (O|(L × M), (Q̂^{∨}, ⊗\overline{P})|S) \ar[r] \ar@{-->}[dr] & ℝHomE′ ; \overline{S}(\overline{P}^{∨}|M, Q^{∨}|L) \\  & ℝHom_{S}^{E′′}(Q̂|L, P |M)} (32)

When passing to cohomology, the first and the last item in this diagram are both equal to Ext_{O_{S}}^{•}(Q, P). We claim that the induced isomorphism on cohomology is equal to the identity. For simplicity, we will prove this for the case that P and Q are the trivial rank one local systems. Then the differential Batalin-Vilkovisky modules of Diagram (32) are invertible. We orient them using σ^{n}, (−1)^{n}σ^{n}, and (θ ⌟ σ^{n})|_{M}, (θ′ ⌟ σ^{n})|_{L} and (θ′′ ⌟ σ^{n})|_{M}, respectively, as in Section 3.3. Then the homomorphisms in Diagram (32) do not preserve orientations according to Definition 1.37, because of the presence of the functions g, defined in (23), entering into the definition of ψ, Equation (24).

Let us call these functions, from the top to the bottom, g_{1}, g_{2}, g_{3}, g_{4}. We also need more detailed notation for the various maps β of Remark 3.9 and introduce

β_{ij} : F^{(i)} → T_{S} ⟶^{⌟ σ} Ω_{S} → F^{(j)}^{∨} ,

where i, j = 0, 1, 2 denotes the number of primes on the letter F. Using similar notation, we introduce the functions

h_{ij} = (θ^{(j)})^{∨} ∘ det β_{ij} ∘ θ^{(i)} .

These are functions on S, invertible where they are defined.

On the submanifold M, we have canonical isomorphisms α_{ij} : F^{(i)}|_{M} → F^{(j)}|_{M} and functions

a_{ij} = (θ^{(j)})^{−1} ∘ det α_{ij} ∘ θ^{(i)} .

Similarly, on L, we have canonical isomorphisms γ_{ij} : F^{(i)}|_{L} → F^{(j)}|_{L} and functions

c_{ij} = (θ^{(j)})^{−1} ∘ det α_{ij} ∘ θ^{(i)} .

With these notations, we now have:

g_{1} = h_{01}a_{01} ,
g_{2} = h_{10}c_{10} ,
g_{3} = h_{12}c_{12} ,
g_{4} = h_{21}a_{21} .

Hence the default of the maps in (32) to preserve orientations is given by the product

\frac{g_{2}((−1)^{n}g_{4})}{g_{1}((−1)^{n}g_{3})} = \frac{h_{10}c_{10}h_{21}a_{21}}{h_{01}a_{01}h_{12}c_{12}} = \frac{c_{10}a_{21}}{a_{01}c_{12}} ,

noting that h_{ij} is dual, and hence equal, to h_{ji}.

Now note that we have two orientations on M, namely (θ ⌟ σ^{n})|_{M} and (θ′′ ⌟ σ^{n})|_{M}. On Ext_{O_{S}}^{•}(O_{L}, O_{M}), this difference induces a factor of a_{20}. Thus, to prove our claim, we need to show that

a_{20} = \frac{c_{10}a_{21}}{a_{01}c_{12}} .

Now, it is clear that a_{ij}a_{jk} = a_{ik}, and c_{ij}c_{jk} = c_{ik}. Thus, our claim is equivalent to

c_{20} = 1 ,

which is true, because c_{02} is the homomorphism of Diagram (31), which is the identity, by assumption. ⊓ ⊔

4.3 Oriented case

Theorem 4.4. Let L and M be immersed Lagrangians of the symplectic manifold S of dimension 2n. Then every orientation of M defines a generator for the bracket of Theorem 4.2. More precisely, every trivialization ω_{M}^{∘} of Ω_{M}^{n} defines a differential d^{∘} on Tor_{O_{S}}^{•}(O_{L}, O_{M}) making the latter a sheaf of Batalin-Vilkovisky algebras.

Proof. From Example 1.36 and Proposition 1.35, we get that ω_{M}^{∘} defines a differential d^{∘} on Tor_{O_{S}}^{•}(O_{L}, O_{M}). We have to show that d^{∘} does not depend on the polarization. For this we repeat the proof of Theorem 4.3, making sure that all morphisms of differential Batalin-Vilkovisky modules preserve orientations. For this, we have to be more careful with our choices. Of course, F and F′′, the two polarizations to be compared, are given. But we will choose F′ in a special way, as follows.

First note that on L, both F |_{L} and F′′|_{L} are complements to T_{L} inside T_{S}|_{L}. Thus we obtain an isomorphism ϕ : F |_{L} → F′′|_{L}, characterized by ϕ(X)−X ∈ T_{L}, for all X ∈ F |_{L}. There exists a canonical subbundle H ⊂ T_{S}|_{L}, such that H is complementary to F |_{L}, F′′|_{L} and T_{L}, and the isomorphism ϕ̃: F |_{L} → F′′|_{L}, characterized by ϕ̃(X)−X ∈ H is equal to −ϕ. (Essentially, H is obtained by negating the F′′-components of the vectors in T_{L}, but preserving their F -components.) The subbundle H ⊂ T_{S}|_{L} is isotropic, so we can extend it, at least locally, to a Lagrangian subbundle F′ ⊂ T_{S}. With this choice we will have

\frac{h_{10}}{h_{12}} = (\frac{h_{10}}{h_{12}} )^{∨} = (−1)^{n}c_{02} ,

and hence

g_{2} = (−1)^{n}g_{3} .

Now, finally, we choose, first an orientation ω_{L}^{∘} of L and then θ and θ′ in such a way that

g_{1}(θ ⌟ σ^{n})|_{M} = ω_{M}^{∘} ,
g_{2}(θ′ ⌟ σ^{n})|_{M} = ω_{L}^{∘} .

Then, by the choice of F′, we have

g_{3}(θ′ ⌟ (−1)^{n}σ^{n})|_{M} = ω_{L}^{∘} .

We choose θ′′ in such a way that c_{02} = 1, as above. Then h_{10} = (−1)^{n}h_{12}, and hence h_{01} = (−1)^{n}h_{21} and h_{01}a_{01}a_{20} = (−1)^{n}h_{21}a_{21}. In other words, g_{1}a_{20} = (−1)^{n}g_{4}, or

g_{4}(θ′′ ⌟ (−1)^{n}σ^{n})|_{M} = g_{1}(θ ⌟ σ^{n})|_{M} = ω_{M}^{∘} .

Now all four homomorphisms of Diagram (32) preserve orientations, and hence they are equal to the morphisms of Diagram (30). This finishes the proof. ⊓ ⊔

Corollary 4.5. In the non-oriented case, the sheaf of Gerstenhaber algebras Tor_{O_{S}}^{•}(O_{L}, O_{M}) is locally a Batalin-Vilkovisky algebra, albeit in a noncanonical way.

4.4 The exchange property

Let S and S′ be a complex symplectic manifolds, of dimension 2n, 2n′, respectively.

Definition 4.6. A symplectic correspondence C → S′ × S is called regular, if for every point P ∈ C one can find polarizations E ⊂ Ω_{S}, defined in a neighborhood of π(P) ∈ S, and E′ ⊂ Ω_{S′}, defined in a neighborhood of π′(P) ∈ S′, such that

(i) C is transverse to E′^{⊥} × E^{⊥} inside S′ × S,

(ii) the induced foliations F, F′ on C descend to foliations F̃, F̃′ on S′ and S, respectively, as in Section 3.2.

Theorem 4.7. Let C → S′ × S be a regular symplectic correspondence. Let L → S be an immersed Lagrangian transverse to C and M′ → S′ an immersed Lagrangian transverse to C. Then there is a canonical isomorphism of sheaves of Gerstenhaber algebras

Tor_{O_{S′}}^{•} (O_{L′}, O_{M′}) = Tor_{O_{S}}^{•}(O_{L}, O_{M}) ,

with notation as in (13).

If L, M′ and C are oriented, then this is an isomorphism of sheaves of Batalin-Vilkovisky algebras.

Proof. We apply the exchange property, Theorem 3.12, twice, first to C → S′ × S, then to S′ → S′ × S′. ⊓ ⊔

Theorem 4.8. Let C → S′ × S be a regular symplectic correspondence. Let L → S be an immersed Lagrangian transverse to C and M′ → S′ an immersed Lagrangian transverse to C. Let P′ be a local system on M′ and Q a local system on L. Then there is a canonical isomorphism of sheaves of Batalin-Vilkovisky modules

Ext_{O_{S′}}^{•} (Q|_{L′}, P′) = Ext_{O_{S}}^{•}(Q, P′|_{M}) ,

covering the isomorphism of sheaves of Gerstenhaber algebras of Theorem 4.7.

Proof. We apply the exchange property, Theorem 3.15, twice, first to C → S′×S, then to S′ → S′×S′. The details are similar to the proof of Theorem 4.3.

⊓ ⊔

An example

Let M be a complex manifold and S = Ω_{M} the cotangent bundle with its canonical symplectic structure. There are two typical examples of immersed Lagrangians:

(i) the graph of a closed 1-form ω ∈ Γ (M, Ω_{M}) which we denote by Γ_{ω} ⊂ Ω_{M}. This is in fact embedded.

(ii) the conormal bundle C_{Z/M} → Ω_{M}, where Z → M is an immersion of complex manifolds, i.e., a holomorphic map, injective on tangent spaces.

Given one of each, we consider the Lagrangian intersection Γ_{ω} ∪ C_{Z/M}. Note that it is supported on Z(ω) ∩ Z. We use the notation

T_{M}(ω, Z) = Tor_{O_{Ω_{M}}}^{•} (O_{Γ_{ω}}, O_{C_{Z/M}}) ,

and

E_{M}(ω, Z) = Ext_{O_{Ω_{M}}}^{•} (O_{Γ_{ω}}, O_{C_{Z/M}}) .

Let f : M → N be a holomorphic map between complex manifolds M, N. Consider the symplectic manifolds S′ = Ω_{M} and S = Ω_{N}. The pullback vector bundle f^{∗}Ω_{N} is then a symplectic correspondence C:

\xymatrix{f^{∗}Ω_{N} \ar[d] \ar[r] & Ω_{N} \\ Ω_{M}}

Let us assume that f^{∗}Ω_{N} → Ω_{M} fits into a short exact sequence of vector bundles

\xymatrix{0 \ar[r] & K \ar[r] & f^{∗}Ω_{N} \ar[r] & Ω_{M} \ar[r] & Ω_{M/N} \ar[r] & 0}.

Then the symplectic correspondence C = f^{∗}Ω_{N} is regular.

If Z → M is an immersion (i.e., injective on tangent spaces), such that the composition Z → N is also an immersion, then the conormal bundle C_{Z/M} → Ω_{M} is an immersed Lagrangian transverse to f^{∗}Ω_{N}. The corresponding immersed Lagrangian of Ω_{N} is the conormal bundle C_{Z/N}.

If ω ∈ Γ (N, Ω_{N}) is a closed 1-form, then its graph is a Lagrangian submanifold of Ω_{N}, which is automatically transverse to f^{∗}Ω_{N}. The corresponding Lagrangian submanifold of Ω_{M} is the graph of the pullback form f ∗ ω.

Corollary 4.9. There is a canonical isomorphism of Gerstenhaber algebras with Batalin-Vilkovisky modules

T_{N}(ω, Z) = T_{M}(f^{∗}ω, Z) , E_{N}(ω, Z) = E_{M}(f^{∗}ω, Z) .

5 Further remarks

5.1 Virtual de Rham cohomology

Let M and L be Lagrangian submanifolds of the complex symplectic manifold S. Let X be their scheme-theoretic intersection. Let E = Ext_{O_{S}}^{•}(O_{L}, O_{M}) be endowed with the differential d from Section 4.2. The sheaf E is a coherent O_{X}-module, the differential d is ℂ-linear.

Definition 5.1. We call (E, d) the virtual de Rham complex of X.

Theorem 5.2. The complex (E, d) is constructible.

Proof. The claim is local in X, so we may assume that the symplectic manifold S is the cotangent bundle of the manifold M, that the first Lagrangian M is the zero section and the second Lagrangian L is the graph of an exact 1-form df, where f : M → ℂ is a holomorphic function. In this form theorem was proved by Kapranov; see the remarks towards the bottom of page 72 in [2]. ⊓ ⊔

Corollary 5.3. The hypercohomology group ℍ^{p}(X, (E, d)) is finite dimensional. Moreover, for Z ⊂ X Zariski closed, ℍ_{Z}^{p}(X, (E, d)) is also finite dimensional.

By abuse of notation, we will write ℍ^{p}(X, E) and ℍ_{Z}^{p}(X, E), instead of ℍ^{p}(X, (E, d)) and ℍ_{Z}^{p}(X, (E, d)), respectively.

Definition 5.4. We call the hypercohomology group ℍ^{p}(X, E) the p-th virtual de Rham cohomology group of the Lagrangian intersection X.

Corollary 5.5. The function

P ⟼ ∑_{i}(−1)^{i} dim_{ℂ} ℍ_{{P }}^{i}(X, E)

is a constructible function χ : X → ℤ.

We may think of χ : X → ℤ as the fiberwise Euler characteristic of the constructible complex (E, d).

5.2 A speculation in Hodge theory

Remark 5.6. There is the standard spectral sequence of hypercohomology

E_{1}^{pq} = H^{q}(X, E^{p}) ⟹ ℍ^{p+q}(X, (E, d)) .  (33)

This should be viewed as a generalization of the Hodge to de Rham spectral sequence.

There is also the usual local do global spectral sequence

E_{2}^{p,q} = H^{p}(X, E^{q}) ⟹ Ext_{O_{S}}^{p+q}(O_{L}, O_{M}) .  (34)

One may speculate to what extent these spectral sequences degenerate.

Example 5.7. For example, if M is a manifold and S = Ω_{M} the cotangent bundle with its standard symplectic structure, and we consider the intersection of M (the zero section) with itself, we get:

Ext_{O_{S}}^{p}(O_{M}, O_{M}) = Ω_{M}^{p} .

Moreover, (E, d) = (Ω_{M}^{•}, d) is the de Rham complex of M and Lagrangian intersection cohomology is equal to de Rham cohomology of M. Thus the spectral sequence (33) is the usual Hodge to de Rham spectral sequence:

E_{1}^{pq} = H^{q}(M, Ω^{p}) ⟹ ℍ^{p+q}(M, (Ω_{M}^{•}, d)) .

On the other hand, we have

Ext_{O_{S}}^{i}(O_{M}, O_{M}) = ⨁_{p+q=i} H^{i}(M, Ω^{q}) ,

in other words, the E_{2}-term of the spectral sequence (34) is equal to the abutment. Thus Ext_{O_{S}}^{i}(O_{M}, O_{M}) is equal to Hodge cohomology of M.

We may, then, rewrite the Hodge to de Rham spectral sequence (33) as

Ext_{O_{S}}^{i}(O_{M}, O_{M}) ⟹ ℍ^{i}(M, (E, d)) .

This, of course, degenerates if M is proper and gives the equality

Ext_{O_{S}}^{i}(O_{M}, O_{M}) = ℍ^{i}(M, (E, d)) ,  (35)

if M is Kähler, by Hodge theory.

The following conjecture is thus a generalization of Hodge theory:

Conjecture 5.8. Under sufficiently strong hypotheses, including certainly that the intersection X = L∩M is complete and some analogue of the Kähler condition, for example that X is projective, we we have

ℍ^{p}(X, (E, d)) = Ext_{O_{S}}^{p}(O_{L}, O_{M}) .

5.3 A differential graded category

Let S be a symplectic variety and 𝔘 = {U_{i}} an affine open cover of S. Construct a category A as follows: objects of A are pairs (M, P), where M is a Lagrangian submanifold of S and P is a local system on M. We do not assume that M → S is a closed immersion, it suffices that this map be affine. We often omit the first component of such a pair (M, P) from the notation.

For objects (M, P) and (L, Q) of A we define Hom_{A}(Q, P) to be the total complex associated to the double complex

C^{•}(𝔘, (E^{•}, d))

given by Cech cochains with respect to the cover 𝔘 with values in the virtual de Rham complex E = Ext_{O_{S}}^{•}(Q, P) (endowed with the differential from Theorem 4.3).

Theorem 5.9. This defines a differential graded category.

Proof. For simplicity of notation, we deal only with Lagrangian submanifolds M, N and L, leaving the generalization to local systems to the reader. There are natural Yoneda pairings (all tensors and Ext’s are over O_{S})

Ext^{i}(O_{M}, O_{N}) ⊗ Ext^{j}(O_{N}, O_{L}) ⟶ Ext^{i+j}(O_{M}, O_{L}) .

We need to show that these are compatible with the canonical differential of Section 4.2. This is a local question, so we may assume that S has three Lagrangian foliations F, F′ and F′′, all transverse to each other, and all transverse to M, N, L. Let s, t′, u′′ be the Euler sections of M, N, L with respect to F, F′ and F′′, respectively. Then we can represent Ext^{•}(O_{M}, O_{N}) as the cohomology of (Ω_{S}^{•}, s − t′), and Ext^{•}(O_{N}, O_{L}) as the cohomology of (Ω_{S}^{•}, t′ − u′′), and Ext^{•}(O_{M}, O_{L}) as the cohomology of (Ω_{S}^{•}, s − u′′). So the claim will follow if we can produce a morphism of complexes

(Ω_{S}^{•}, s − t′) ⊗ (Ω_{S}^{•}, t′ − u′′) ⟶ (Ω_{S}^{•}, s − u′′) .

But this is easy: just take the cup product. ⊓ ⊔

Remark 5.10. The cohomology groups of the hom-spaces in A are the virtual de Rham cohomology groups of Lagrangian intersections.

Remark 5.11. The category A does not depend on the affine cover 𝔘 in any essential way.

Remark 5.12. Of course, it is tempting to speculate on relations of A to the Fukaya category of S. We will leave this to future research.

5.4 Relation to vanishing cycles

Let S be a complex symplectic manifold of dimension 2n. Let L, M Lagrangian submanifolds, and X = L ∩ M their intersection.

In [1], we introduced for any scheme X a constructible function ν_{X} : X → ℤ. The value ν_{X}(P) is an invariant of the singularity (X, P). In our context, the singularity (X, P) is the critical set of a holomorphic function f : M → ℂ, locally defined near P ∈ M. Hence (see [1]), the invariant ν_{X}(P) is equal to the Milnor number of f at P, i.e., we have

ν_{X}(P) = (−1)^{n}(1 − χ(F_{P})) ,

where F_{P} is the Milnor fibre of f at the point P.

Conjecture 5.13. We have χ(P) = ν_{X}(P).

This conjecture would follow from Remark 2.12 (b) of [2]. Note that Kapranov refers to this as a fact, which is not obvious, although probably not very difficult.

Conjecture 5.14. If the intersection X is compact, so that the intersection number #^{vir}(X) is well-defined, we have

#^{vir}(X) = ∑_{i}(−1)^{i} dim ℍ^{i}(X, E) ,

i.e., the intersection number if equal to the virtual Euler characteristic of X, defined in terms of virtual de Rham cohomology.

To see that Conjecture 5.13 implies Conjecture 5.14, recall from [1], that the intersection X has a symmetric obstruction theory. The main result of [1] implies that #^{vir}(X) = χ(X, ν_{X}). But if χ = ν_{X}, then χ(X, ν_{X}) = χ(X, χ) = ∑_{i}(−1)^{i} dim ℍ^{i}(X, E).

Remark 5.15. If S is the cotangent bundle of M, and L is the graph of df, where f : M → ℂ is a holomorphic function, then the Lagrangian intersection X = L ∩ M is the critical set of f. Thus X carries the perverse sheaf of vanishing cycles Φ_{f}. In [2], Kapranov constructs, at least conjecturally, a spectral sequence whose E_{2}-term is (E, d) and whose abutment is, in some sense, Φ_{f}.

Conjecture 5.16. In the general case of a Lagrangian intersection X = L ∩ M inside a complex symplectic manifold S, we conjecture the existence of a natural perverse sheaf on X, which locally coincides with the perverse sheaf of vanishing cycles of Remark 5.15. There should be a spectral sequence relating (E, d) to this perverse sheaf of vanishing cycles. We believe that [3] may be related to this question. This conjecture, in some sense, categorifies Conjecture 5.13.

References

1. K. Behrend, Donaldson-Thomas invariants via microlocal geometry,

   math.AG/0507523, July 2005.

2. M. Kapranov, On DG-modules over the de Rham complex and the vanishing cycles functor, Algebraic Geometry (Chicago, 1989), Lecture Notes in Math., 1479, Springer, Berlin, 1991, 57–86.

3. M. Kashiwara, P. Schapira, Constructibility and duality for simple holonomic modules on complex symplectic manifolds, math.QA/0512047, December 2005.

4. D. Tamarkin, B. Tsygan, The ring of differential operators on forms in noncommutative calculus, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc., Providence, RI, 2005, 105–131.